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KINEMATICS OF MACHINES 



AN ELEMENTARY TEXT-BOOK 






BY 



R.7. pURLEY, B.Sc, Ma.E. 

Thomas Workman Professor of Mechanical Engineering^ 
McGill University , Montreal 



FIRST EDITION 
FIRST THOUSAND 



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> 3 J 5 3 



NEW YORK 

JOHN WILEY & SONS 

London: CHAPMAN & HALL, Limited 

1903 ■ 






THE LIBRARY OF 
CONGRESS, 

Two Copies Received 

FEB 4 1903 

Copyiight Entry 

«^ XXo. No. 
COPY B. 



LASS 



Copyright, 1903, 

BY 

R. J. DURLEY. 






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ROBERT DRUMMOND, PRINTER, NEW YORK. 



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PREFACE. 



This book is intended as a brief manual for Engineer- 
ing Students, and treats^ chiefly of those portions of the 
subject of the Kinematics of Machines which are likely to 
be of assistance in the study of the Dynamics of Machines 
and in work in Machine Design. 

The author wishes to thank his friends and colleagues, 
Dr. E. G. Coker and Mr. H. M. Jaquays, for their kindly 
criticism and for their help in revision of the proof-sheets. 

Many earlier works have been consulted in the prepara- 
tion of this volume ; wherever possible they are named in 
the text or in foot-notes. Professors John H. Barr and 
C. W. MacCord have courteously permitted the use of cer- 
tain of their diagrams, and the author is indebted to 
The American Stoker Company, The Brown and Sharpe 
Manufacturing Company, and The Link Belt Engineering 
Company for the use of figures and information. 

Montreal, November 1902. 

iii 



CONTENTS. 



CHAPTER I. 



Introductory Considerations. 

SBC. PACK 

1 . Study of Machines i 

2. Constrained Motion 3 

3. Pairs of Elements 3 

4. Links and Chains ^ 6 

5. Motion and Position in a Plane 9 

6. Non-plane Motion 17 

7. Freedom and Constraint 19 

8. Elements and Pairs in Rigid Links 21 

9. Pairing of Non-rigid Links 24 

ID. Classification of Mechanisms 25 

CHAPTER II. 
Position, Velocity, and Acceleration. ' 

11. Velocity 27 

12. Uniform Velocity 28 

13. Variable Velocity 29 

14. Uniform Acceleration 31 

15. Acceleration in General 33 

16. Composition of Velocities and Accelerations 35 

17. Resultant Acceleration 37 

18. Diagrams of Displacement and Velocity 38 

19. Diagrams of Acceleration 44 

20. Diagrams on a Displacement Base 49 

21. Acceleration Diagrams on a Displacement Base 51 

22. Polar Diagrams of Displacement, Velocity, and Acceleration 53 

23. Diagrams for Simple Harmonic Motion 58 

24. Relative Motion of Two Bodies Each having Simple Harmonic Motion. . . 63 

25. Composition of Simple Harmonic Motion not along Same Line 67 

V 



VI CONTENTS. 

CHAPTER III. 
Plane Mechanisms Containing only Turning Pairs. 

SBC. rXGB 

26. Quadric Crank-cliains 70 

27. Virtual Centres and Centrodes 71 

28. Angular Velocities 73 

29. Inversions of the Quadric Crank-chain 76 

30. Change Points and Dead Points 81 

31. Special Forms of Quadric Crank-chain 83 

32. Straight-line Motions 87 

33. Accurate Straight-line Motions 90 

, CHAPTER IV. 
Slider-crank Chains. 

34. Slider-crank Chains 97 

35. Velocity and Acceleration of Cross-head in Direct-acting Engine 99 

36. Graphic Methods for Cross-head Velocity and Acceleration 103 

37. Angular Velocity and Acceleration of Connecting-rod no 

38. Angular Velocity of Cylinder in Oscillating Engine 112 

39. Whitworth Quick-return Motion 118 

40. Pendulum Pump 120 

41. Crossed Slider-crank Chains 122 

42. Double Slider-crank Chain 123 

43. Elliptic Trammels 126 

44. Oldham's Coupling 127 

45. Crossed-slide Chains 131 

46. Straight-line Motions Derived from Slider-crank Chains 136 

47. Chain Containing Sliding Pairs Only 138 

CHAPTER V. 

Determination of Velocity and Acceleration in Plane Mechanisms. 

48. Velocity and Acceleration Determined from Virtual Centres 141 

49. Method by Using Point-paths 143 

50. Polar Diagrams of Velocities for Simple Plane Mechanisms 146 

51. Indirect Method in More Complex Cages 152 

52. Polar Acceleration Diagrams for Plane Mechanisms 155 

53. Example of Polar Velocity and Acceleration Diagrams 160 

CHAPTER VI. 
Alteration of Mechanisms. Closure. 

54. Expansion of Elements 164 

55. Augmentation of Chains 166 



CONTENTS, vu 



SEC. 



PAGB 



56. Reduction of Chains 168 

57. Reduction by Use of Centrodes 17° 

58. Closure of Incomplete Pairs ^7^ 

59. Closure of Incomplete Chains 172 

CHAPTER VII. 

Constraint and Velocity Ratio in Higher Pairing Involving Plane 

Motion. 

60. Restraint of Bodies having Plane Motion. i77 

61. Closed Higher Pairs having Plane Motion 183 

62. Form of Elements for a Given Motion 186 

63. Condition for Uniform Velocity Ratio i88 

64. Wheel-gearing ^9^ 

65. Spur-w^heels ^93 

66. Involute Teeth ^95 

67. Cycloidal Teeth . • . I97 

CHAPTER VIII. 
Wheel-trains and Mechanisms containing them. Cams. 

68. Simple and Compound Wheel-trains 201 

69. Epicyclic Gearing 2°5 

70. Mechanisms containing Wheel-trains 209 

71. Cam-trains ^^3 

72. Rotating Cams ^^5 

73. Sliding and Cylindrical Cams 219 

74. Velocity Ratio in Cam-trains 222 

CHAPTER IX. 

Ratchet Mechanisms and Escapements. 

75. Ratchet-gearing ^^7 

76. Running Ratchets 228 

77. Stationary, Checking, and Releasing Ratchets 230 

78. Escapements ^37 

CHAPTER X. 

Mechanisms Involving Non-rigid Links. 

79. Non-rigid Links ^44 

80. Velocity Ratio in Belt-gearing. Length of Belts 244 

81 Belt-gearing for Variable Velocity Ratio 246 

82. Velocity Ratio in Chain and Rope-gearing 251 

83. Belt- and Rope-gearing between Non-parallel Axes 254 



viil CONTENTS. 

SEC. PAGE 

84. Springs 256 

85. Fluid Links and Pressure Pairs „ 259 

%6. Chamber Crank-trains 261 

87. Chamber Wheel-trains 264 

88. Ratchet-trains containing Non-rigid Links 268 

89. Pressure Escapements containing Fluid Links 273 

CHAPTER XI. 
Chains Involving Screw Motion. 

90. Formation of Screw Surfaces 276 

91. Screw Mechanisms Involving Lower Pairing of Rigid Links 278 

92. Screw Mechanisms containing Fluid Links 282 

93. Screw-wheels and Worm-gearing » . 285 

94. Forms of Teeth in Screw-gearing and Worm-gearing 293 

95. Hyperboloidal Wheels 298 

CHAPTER XIL 
Spheric Motion. 

96. Spheric Motion in General 304 

97. Spheric Mechanisms having Lower Pairing ; the Conic Quadric Crank-chain 308 

98. Spheric Mechanisms having Higher Pairing; Bevel Gear 317 

99. Roller Bearings Involving Spheric Motion. ...» 33 1 

100. Ball-bearings 334 

CHAPTER XIIL 
Kinematic Classification of Mechanisms. 

101. Historical Sketch 346 

102. Classification of Willis ; Babbage's Notation 349 

103. Classification and Notation of Reuleaux 351 

104. Classification of Hearson 354 

105 . Remarks on Classification 356 



KINEMATICS OF MACHINES. 



CHAPTER I. 
INTRODUCTORY CONSIDERATIONS. 

I. Study of Machines. — In general the study of a 
Machine involves problems of three distinct kinds. We 
may first of all consider from a geometrical point of view 
the motion of any part of the machine with reference to 
any other part, without taking account of any of the forces 
acting on such parts. Or, the action of the forces impressed 
on the parts of the machine, and of the forces due to its 
own inertia or to the weight of its parts, may be dealt with, 
and the resulting transformations of energy may be deter- 
mined. A third branch of the theory of machines treats 
of the action of these loads and forces in producing stresses 
and strains in the materials employed in the construction 
of the machine, and discusses the sizes, forms, and pro- 
portions of the various parts which are required either to 
insure proper strength while avoiding waste of material, 
or to make the machine capable of doing the work for which 
it is being designed. 

The science dealing with the first-named class of problem 
is termed the Kinematics of Machines, which we may define 
as being that science which treats of the relative motion 
of the parts of machines, without regard to the forces pro- 
ducing such motions, or to the stresses and strains produced 
by such forces. 



2 KINEMATICS OF MACHINES, 

With this limitation, in the case of almost all bodies 
forming portions of machines, it is possible to neglect any 
deformation they may undergo in working, and in studying 
the Kinematics of Machines we may at once apply to 
machine problems the results obtained by the study of 
the motion of rigid bodies. Important exceptions will 
present themselves to the reader's mind; for example, 
ropes, belts, and springs cannot be considered kinematically 
as being rigid, and many mechanical contrivances involve 
the use of liquid or gaseous material. Such cases as these 
will be considered later. 

By the term Machine we may understand a combina- 
tion or arrangement of certain portions of resistant material, 
the relative motions of which are controlled in such a way 
that some form of available energy is transmitted from 
place to place, or is transformed into another desired kind. 
This definition includes under the head of Machines all 
contrivances which have for their object the transformation 
or transmission of energy, or the performance of some par- 
ticular kind of work, and further implies that a single 
portion of material is not considered as a machiae. The 
so-called simple machines in every case involve the idea of 
more than one piece of material. 

A combination or arrangement of portions of material by 
means of which forces are transmitted or loads are carried 
without sensible relative motions of the component parts 
is called a Structure. 

The term Mech,anism is often used as an equivalent for 
the word Machine. It is, however, preferable to restrict 
its use somewhat, and to employ the word to denote simply 
a combination of pieces of material having definite relative 
motions, one of the pieces being regarded as fixed in space. 
Such a mechanism often represents kinematically some 
actual machine which has the same number of parts as the 
mechanism with the same relative motions. The essential 
difference is that in the case of a machine such parts have 



INTRODUCTORY CONSIDERATIONS. 3 

to transmit or transform energy, and are proportioned and 
formed for this end, while in a mechanism the relative motion 
of the parts only is considered. We may look upon a 
mechanism, then, as being the ideal or kinematic form of a 
machine, and our work will be much simplified in most 
cases if we consider for kinematic purposes the mechanism 
instead of the machine. Such a substitution is also of the 
greatest service in the comparison and classification of 
machines ; we shall find in this way that machines, at first 
sight quite distinct, are really related, inasmuch as their 
representative mechanisms consist of the same number of 
parts having similar relative motions, and only differing 
because a different piece is considered to be fixed in each 
case. 

2. Constrained Motion. — On further consideration of the 
nature of a Machine as defined above, it will be noted that 
each part of the machine must have certain definite motions 
relatively to any other part, such definite motions being 
repeated again and again during the working of the machine. 
Thus the motion of a machine-part must be completely 
constrained, that is, the part must be free to move only in 
the manner desired to produce the required transformation 
of energy, and for it other unnecessary motions must be 
rendered impossible. Constrained motion of a body takes 
place when every point in the body is made to describe some 
definite and prescribed path. This constraint is effected 
in general by so forming and connecting the parts that all 
forces tending to disturb their constrained motion are 
balanced by stresses set up in the parts themselves. It is 
assumed, of course, that the machine remains uninjured 
by such stresses. 

3. Pairs of Elements. — The nature of the connection 
between the parts of a machine will be best understood by 
taking a simple case and discussing the way in which some 
form of constrained relative motion of two bodies' may be 
obtained. Suppose, for example, that a piece of material, 



4 KINEMATICS OF MACHINES. 

which we may call a, has to be capable of a motion of 
translation along a straight line, with reference to another 
piece, h, and is to have no other relative motion whatever. 
This must be accomplished by giving these pieces suitable 
forms. Such an arrangement as that sketched in Fig. i 




Fig. I. 



would not meet the case, for, although a executes the re- 
quired movement so long as it remains in the groove formed 
in h and does not rotate on its axis in the groove, the forms 
shown do not prevent a leaving the groove in 6 or rotating 
in that groove. 

It will be found that to attain the desired object some 
such forms as shown in Fig. 2 must be adopted, and that if 
this is done, the only possible motion of a relatively to h is 
that of simple translation along a straight line parallel to 
the edge of the groove or slot in h. The figure will recall 
to the reader the appearance of a steam-engine cross-head 
and its guides, a pair of bodies which have indeed the same 
relative motion as that described above. 

We shall refer to a pair of bodies so formed as to permit 



INTRODUCTORY CONSIDERATIONS. 5 

of partly or wholly constrained relative motion while in con- 
tact as a pair of elements, the elements being really the sur- 
faces of contact, or working surfaces, of the pair of bodies. 
Such pairs are distinguished as being (a) higher pairs and 
{h) lower pairs. Lower pairs may be defined as those in 
which ''the forms of the elements are geometrically iden- 
tical, the one being solid or full and the other hollow or 
open" (Reuleaux). This definition involves the idea of 
surface contact to produce the required partial or complete 
constraint, while in the case of higher pairs constraint is 
produced by contact at a sufficient number of lines or 




Fig. 2. 



points. Mechanically, lower pairing in machinery is pref- 
erable, wherever possible. The reason for this is that 
wear takes place much more rapidly in a case where line 
or point contact occurs than in the case where surfaces of 
considerable extent are touching, other conditions being 
the same. 

A pair of elements whose relative motion is completely 
constrained is said to be closed. Thus such a pair as is 
shown in Fig. i is not closed, while that of Fig. 2 is com- 
pletely closed; for, as has been already pointed out, the 
only possible relative motion is one of pure translation in 
a straight line. 



6 KINEMATICS OF MACHINES. 

The nature of the relative motion of two bodies can only 
be defined when one of them is considered as being fixed. 
In the case of a pair of elements ah, a being fixed while h 
moves, we may have the same relative motion of a and h 
as when h is fixed while a moves, but the pair is said to be 
inverted, that is, the second element is fixed instead of the 
first. Examples of such inversion of pairs frequently occur 
in considering actual machines, and it is important to remem- 
ber that, while inversion of a pair may cause no alteration 
of the relative motion of the elements themselves, it may, and 
generally does, alter their motion relatively to other bodies. 

4. Links and Chains. — In studying any simple mechan- 
ism or machine, we find that each piece of material carries, 
or has formed upon it, one element of each of two or more 
pairs. Take for example the cross-head of a steam-engine ; 
in addition to the surface which pairs with the guide bar or 
bars, the block has a cylindrical surface pairing with a similar 
one on the small end of the connecting-rod, and it thus carries, 
or links together, two elements belonging to two different 
pairs. 

In general, then, a part of a machine forms a kinematic 
link connecting two or more elements, belonging respectively 
to two or more pairs, and the whole arrangement or com- 
bination of such links is known as a kinematic chain. This 
may or may not have such kinematic properties as to make 
it available as a mechanism; for we can easily imagine a 
kinematic chain which does not comply with our definition 
of a mechanism when one link is fixed. Consider the case 
of a linkwork formed of five bars, ah c d e, jointed at the 
angles as shown in Fig. 3. Suppose a to be fixed, then the 
motion oi c or d relatively to a is not constrained, and such 
a chain, therefore, is not a mechanism as we have defined it. 

It is most important to note, with regard to this point, 
that the motion of c with respect to h is constrained, i.e., c 
can only have one motion with regard to 6, that of turning 
about the axis of the joint connecting them, whereas with 



INTRODUCTORY COhlSIDERATIONS. 7 

respect to a, c can be made to move in any number of dif- 
ferent ways, depending in this case on the force or forces 
appUed to the different bars. The motion of c with respect 
to a is therefore not constrained. This fact is illustrated 
in Fig. 3, where it is seen that if the links h and e take up 
the positions h' and e\ c and d may be either at c' and d\ or 
at c'^ and d'\ Such a kinematic chain as this is said not 
to be closed, and we define a closed chain as a series of links 




Fig. 3. 

so connected that each of them has only one definite motion 
relatively to any other link. Thus if one link be fixed, the 
motion of any other can be determined. A closed chain 
having one link fixed is then equivalent to a mechanism. 

The various ways in which closure is obtained in pairs 
and in chains will be discussed later. 

A chain of which each link carries two elements is 
termed a simple chain, for a link cannot have a less ntimber 
of elements than two. If, however, any link or links have 
three or more elements respectively belonging to three or 
more pairs, the chain is said to be compound. In some 
ways compound chains present more difficulties than do 
simple chains, but the kinematics of both kinds may be 



KINEMATICS OF MACHINES. 



s udied by exactly the same methods. Fig. 4 shows a 
cosed compound chain, which has been suggested as a 
straight-line motion. It will be seen that the link a is 
faxed, and that b carries one element of each of the pairs 




Fig. 4. 



ba, hd, be, while c has upon it one element of each of the 
pairs ca, cd, cf. 

It is worth while noticing that if the link d were removed 
the Cham would no longer be a closed one. The particular 
mechanism shown in Fig. 4 will be again referred to * 

In the last two figures the links have been represented 
by straight bars. From a kinematic point of view how- 
ever, the mechanisms or chains would have been unchanged 
if the form of the bars had been altered in any way, always 
supposing that the axes of the joints remain parallel and at 
the same distance apart, and that the forms of the links 
are not such as to cause fouling or interference while the 
mechanism is in motion. It is evident that these re- 
marks apply generally, and wemay say that, as a rule, 

* See Fig. 59. 



INTRODUCTORY CONSIDERATIONS. 9 

the form or shape of a Hnk in a chain is not of importance 
in kinematics, so long as the form adopted does not render 
impossible any portion of the required movement of the 
link. Questions of form and shape fall within the province 
of the science of Machine Design. , 

We have already seen that in discussing whether a 
kinematic chain is or is not equivalent to a mechanism, 
we suppose one link to be fixed, and we then proceed to 
determine whether the chain is closed or not ; a closed chain 
having one link fixed being regarded as a mechanism. 

The choice of the fixed link is left open, and by selecting 
different links of a kinematic chain different mechanisms are 
generally obtained. Thus, in general, from a given kine- 
matic chain we may derive as many mechanisms as the 
chain has links. These mechanisms are called the inver- 
sions of the original chain, and, as in the case of the inver- 
sion of pairs, the exchange of one fixed link for another is 
known as the inversion of the chain. Many examples of 
such inversion will be met with in the following chapters. 

5. Motion and Position in a Plane. — Kinematics is sim- 
ply the science of pure motion, as is indeed indicated by its 
name (from Kiv-qyia^ motion), first suggested by Ampere. 
Some of the simpler propositions of pure kinematics will be 
given here before explaining their application in the special 
case of the kinematics of machines. They are based on 
geometrical principles, since they deal with the ideas of 
position and space. But it will be at once seen that the 
introduction of the ideas of time, and consequently of veloc- 
ity and acceleration, extends the scope of the science of 
kinematics considerably beyond the limits of pure geometry. 

Two chief classes of problems arise, the first dealing with 
the position and motion of a particle, and the second treat- 
ing of similar questions relating to rigid bodies. The motion 
of non-rigid bodies is of course of a far more complex nature^ 
and only a few elementary cases will fall within the limits 
of this work. Indeed the motion of such bodies cannot be 



lo KINEMATICS OF MACHINES. 

investigated apart from the forces acting on them, and its 
consideration falls within the province of Kinetics, rather 
than within that of Kinematics. 

Motion is defined as change of position, and is known 
if the position of the. point or body considered is known for 
every instant. The position of a point or of a body can only 
be defined in relation to another point or body (as the case 
may be) whose position is fixed, or, in other words, whose 
change of position is neglected. Position (and therefore 
motion) is then purely relative. When we speak of a moun- 
tain being ten thousand feet in height, we are referring the 
position of its summit to an arbitrary datum surface, that 
of mean sea-level. In stating the position of a point or 
body (a body being equivalent to a system of points) we 
must then refer to some other point or body, and in consid- 
ering the motion of a point or system of points, such motion 
can only be imagined with reference to a second point or 
system of points, supposed to be fixed. 

In the case of plane motion, this reference system is 
usually taken to be the surface on which is drawn the dia- 
gram representing the motion of the body considered. In 
order to define the plane motion of a plane figure, with 
regard to a plane, it is sufficient to know the motion of any 
two points in the figure with reference to the plane. The 
truth of this will be seen by considering that if the motion 
of one point only were known, we should still be ignorant 
of any rotation the figure might have about an axis perpen- 
dicular to the plane. The knowledge of another point's 
motion, however, defines such rotation. 

In most cases, problems arising in the kinematic study 
of machines are found to involve the consideration of Plane 
Motion only. 

A rigid body having Plane Motion moves in such a way 
that all planes originally parallel to a certain fixed plane 
(that of motion) remain parallel thereto during the whole 
movement of the body, while any point whatever in the body 



INTRODUCTORY CONSIDERATIONS. H 

moves in a plane either parallel to or coincident with the 
plane of motion. 

A body moving in this manner will in fact have no 
motion of translation in a direction normal to the plane, of 
motion, and the position of the body with respect to the 
plane of motion will agree exactly with the position of its 
projection on the plane of motion. Hence in considering 
the plane motion of rigid bodies, we need deal only with the 
kinematics of plane figures, and all propositions relating to 
the plane motion of plane figures will be applicable to that 
of rigid bodies. 

It is not, in general, so necessary to trace out the whole 
motion of a body as to know what is its instantaneous 
motion at some given stage of its movement. By this term 
is meant the change of position executed by the body in a 
very small period of time. The manner in which these 
small changes of position follow one another must now be 
considered for the case of plane motion. 

In Figure 5, let A 5, A'B\ represent two successive posi- 
tions of a plane figure (as defined by the position of two 




I ,'C2 '/- 



Fig. 5. 

points A and B in it) at the beginning and end of an interval 
of time which is very small as compared with the whole 
period of motion. 

Join AA', 5J5', and bisect the lines ylA', 55', by straight 
lines perpendicular lo AA' , BB\ and intersecting at C^, 



12 KINEMATICS OF MACHINES. 

Then it is plain that C^A = CJ\.' and C^B = C^B\ and if 
the point A had described a very small circular arc with 
centre Q, its new position would have been A\ and its path 
would have been indistinguishable from the line AA\ The 
actual infinitesimally small change of position of the point A 
is therefore the same as if it had been rotated in the plane 
of motion around an axis perpendicular to the plane and 
passing through C^, and similarly for the point B. Thus,, 
knowing the change of position of two points in the rigid, 
figure considered, we say that the actual instantaneous motion 
of the hodyAB has been equivalent to a virtual rotation about 
the centre C^. During the next instant the instantaneous 
motion may be around some other point C^ indefinitely 
near to C^, and so on. The point C^ corresponds to the 
movement from A^B^ to A''B'\ Thus to every part of the 
motion oi AB, with regard to the plane, there corresponds 
a certain point C in the plane, about which an equivalent 
virtual rotation has taken place. Such points, as C,^, C2 . . . , 
are called the instantaneous or virtual centres oi AB with 
regard to the plane. The locus of C^, or the curve described 
by the point C on the plane, is known as the centrode oi AB 
with regard to the plane, and, in general, it forms a con- 
tinuous curve. 

In the case of a rigid body having plane motion, it would 
be more correct to consider the equivalent rotation as tak- 
ing place about a virtual axis (perpendicular to the plane of 
motion) of which the points C^, C^ . . . are the successive 
traces on the plane of motion. Such a virtual axis would 
then describe a surface in space, this surface being known 
as the axode of the body with regard to the plane of motion. 
For most cases of plane motion, however, we are content to 
simplify matters by considering the centrode instead of the 
axode. We shall see later that in more complex forms of 
motion the axode becomes of great kinematic importance. It 
is in every case what is called a ruled surface, i.e., a surface 
described by successive positions of a straight line in space. 



INTRODUCTORY CONSIDERATIONS, 13 

Referring again to the plane motion of the figure AB 
{Fig. 5), let us inquire what happens if our construction 
fails. This will occur if the bisectors of the lines A A' and BB' 
are parallel, in which case the successive positions of AB 
are also parallel to one another, and the motion of the body, 
or of the figure it represents, is one of simple translation 
in a straight line. The virtual centre for such motion as 
this is then at an infinite distance, and we may regard any 
plane motion of translation in a straight line as equivalent 
to a rotation about an infinitely distant centre. Again, sup- 
pose that one of our reference points A does not change its 
position at all. It is easily seen that AB has now simple 
rotation about A , and during the continuation of this motion 
we have no longer a virtual but a permanent centre. It 
may happen that the lines bisecting A A' and BB' are coin- 
cident. A little consideration will show that in this case, 
since the triangles ABC^ and A'B'C^ must be equal in all 
respects, the point C^ is at the intersection of AB and A'B\ 
produced if necessary; as before, a simple rotation about 
Cj would suffice to move AB into the new position A'B\ 

It is thus shown that in every case the motion of a plane 
figure in a plane may he regarded as equivalent to a simple 
rotation about some actual or virtual centre, whose position in 
the plane will be fixed in the case of simple rotation, or will 
be at an infinite distance in the case of simple translation. 
Such a virtual centre, however, is in general neither fixed, nor 
at an infinite distance, but changes its position as the body 
moves, and its locus in the plane is the centrode of the body 
with reference to the plane. Note that only rigid bodies 
or figures can have centrodes, for we assume that the posi- 
tion of our reference-line A 5 in the figure or body remains 
unchanged throughout the motion, and we represent a rigid 
body by the line joining the two points in question. 

It has thus been seen that the centrode of a body with 
regard to the plane of motion is a curve described on that 
plane by the virtual centre of the body. Let us now con- 



14 KINEMATICS OF MACHINES, 

sider the relative motion of two bodies in a plane. Instead 
of supposing that the virtual centre M (Fig. 6) of the first 
ho&y AB traces its centrode on the plane of motion, imagine 
that the curve is marked on a sheet of paper or surface rigidly 
attached to the second body CD, and that the body CD is 
fixed. The point M is then the one point common to the 
two hod\&s AB and CD at which there is no relative motion, for 
the only possible relative motion would be rotation about 
the point M, a motion which is non-existent as far as a 
point is concerned. M is the virtual centre oiAB relatively 
to CD, but evidently it might equally well be called the 
virtual centre of CD relatively to AB. Next suppose that 
ABis> fixed, and let CD have exactly the same relative motion 
as before. At the instant when the relative positions of 
AB and CD are the same as those just considered, the vir- 
tual centre will be the same point M, but it may now be 
supposed to describe its centrode on the body AB, and not 
on CD. This centrode (that of CD relatively to AB) will 
not be the same curve as that described before, although 
they must have one point M in common at any instant. It 
is evident, therefore, that the two centrodes corresponding 
to the relative motion of two bodies always touch at a point, 
which is the virtual centre for the instant considered, and 
we may represent such relative motion by the rolling on one 
another of a pair of centrodes. Further, we shall find that 
from the form of these centrodes we can determine the 
relative motion of the two bodies. 

To make this clearer, the two cases of motion are repre- 
sented in Fig. 6. AB and CD represent the original posi- 
tions of the two bodies, and, CD remaining fixed, A^B^, 
AJB^ . . . A.^B^ represent successive positions oi AB, the 
motion from AB to A^B^ corresponding to a rotation about 
a virtual centre M^, and so on. The curve MJVL^ . . . M^ is 
then the centrode oi AB with regard to CD. 

Next we have plotted the positions CJ)^, Cfi^ . . . Cfl^ 
which CD would occupy, supposing that the relative motion 



IN TRODUCTOR Y CONSIDERS TIONS. 



15 



were the same as before, but that AB now remained fixed. 
For example, in the figure CJ)^ has the same position rela- 
tive to AB that A 3^3 has to CD, and so on for all the other 
positions. We now find the series of virtual centres M^ 




Fig. 6. 

N^ . . . N^ by the construction previously explained, and 
see that these centres lie on another curve touching the first 
at Mj and forming the centrode of CD with regard to AB. 

Remembering that this curve is attached to, or rather 
described on, the body represented hy AB, suppose that CD 
remains fixed, while AB (with the centrode attached) moves 
from AB to A^B^^, i.e., AB rotates instantaneously about 
Mj. If the movement is imitated by tracing AB and the 
curve Mj, N2 - - - N^on paper and placing AB in the position 
A 2^2, it will be found that N^ coincides with M^. When 
AB is at ^3^3, M3 and N^ coincide, and so on. Such suc- 
cessive coincidences can only occur if the curve MJSl^ rolls 
on the curve MJV[^, 

In the same way if we trace CD and the curve M^Mg, and 
let CD occupy its successive positions, we find that the points 
coincide as before, the curve MJV[^ now rolling on MJSl^. 

Thus the given relative motion oi AB and CD, through 
the successive positions shown on the figure, is represented 
by the rolling on one another of two curves, the pair of cen- 
trodes of the two bodies. 



1 6 KINEMATICS OF MACHINES. 

The reader is strongly recommended to satisfy himself 
of the correctness of the above statements by actually draw- 
ing a pair of bodies, and their centrodes for a given case of 
relative motion. Great care and accuracy in drawing are 
necessary in order to obtain correct positions for the virtual 
centres. 

We have now discussed the case of the relative motion of 
two bodies in a plane, and have seen that their virtual 
centre describes a pair of curves, namely the centrodes, each 
being traced on one of the two bodies. 

Suppose next that we have three bodies, represented, 
as before, by plane figures, and having any kind of relative 
plane motion. The three bodies will evidently have three 
virtual centres, while four bodies would have six, and so on ; 
in fact, a kinematic chain having plane motion and con- 

Tl (ft — I ) 

sisting of n links will have — ^ virtual centres connected 

2 

with it, for it will easily be seen that the number of virtual 
centres must be that of the combinations of n things taken 
two at a time. 

On examination of any particular case we shall see that 
the various virtual centres in a mechanism having plane 
motion are arranged in threes, each three lying in a straight 
line, whatever be the position of the mechanism. 

The proof of this statement is as follows : Consider any 
three of the bodies, or links forming the kinematic chain 
or mechanism, and let us call them a, b, c. Denoting the 
virtual centre of a with regard to 6 by 0^^ and remembering 
that this is the same point as the virtual centre of b with 
regard to a, we have for the three bodies considered the 
three virtual centres 0^^, 0^^, 0^^. First consider b as being 
fixed. Then with regard to the point 0^^ any point in a 
has a simple motion of rotation, so that, for example, the 
point 0^^ is moving instantaneously and relatively to 6 in a 
direction at right angles to the line 0^^ . . . 0^^. 

Again, with regard to the point 0^^, any point in c, such 



INTRODUCTORY CONSIDERATIONS. 



17 



as 0^^, must be moving instantaneously and relatively to h 
in a direction at right angles to the line 0^^ . . . 0^,^. 

Thus the point 0^^, regarded as a point in a, is moving 
in a line perpendicular to 0^^ ... 0^^; while if regarded as 
a point in c, 0^^ moves in a line perpendicular to 0^^ . . . 0^^, 
h being regarded as fixed in each case. 0^^ cannot have 
two separate directions of instantaneous movement at the 
same instant, hence the lines 0^^ . . . 0^^ andO^^ . . . O^^are 
both perpendicular to the same line. They cannot be 




Fig. 7. 



parallel, since they both pass through 0^^, and they there- 
fore coincide in direction, i.e., the points 0^^, 0^^, 0^^ lie on 
one straight line. 

The position of the virtual centres in various mechanisms 
will be studied when we consider the relative velocities of 
their different parts. In many instances the proposition 
just given is of great assistance in determining the posi- 
tions of the virtual centres in a mechanism. 

6. Non-plane Motion. — In the majority of cases it will 
be found that the relative motions of the parts of machines 
are plane motions, either of rotation or translation, or both 
combined. Such motions can be studied geometrically by 



1 8 KINEMATICS OF MACHINES. 

the method indicated in the preceding section. It is pos- 
sible (as will be seen later) to have a lower pair, in which 
the motion is non-plane. A somewhat limited number of 
cases of higher pairing also occur in which the motion is 
non-plane. 

In every instance, however, in a closed pair, we have seen 
that there must be continuous contact of the surfaces, and 
it follows that the most general possible relative motion of 
two parts of a mechanism is represented by the motion of 
one rigid body continuously touching another at a point 
or series of points. 

Any such motion must be of the nature of sliding, roll- 
ing, or spinning, separately or combined. 

Simple rolling takes place if the instantaneous axis lies 
in the common tangent plane at the point of instantaneous 
contact. 

Simple spinning exists when the instantaneous axis is 
the common normal at the point of contact. 

Suppose that the relative motion is such that the instan- 
taneous axis passes through the point of contact, and is 
neither in nor perpendicular to the tangent plane. The 
motion is then combined rolling and spinning. If the in- 
stantaneous axis does not pass through the point of contact, 
the rolling and spinning will further be combined with a 
sliding motion. 

We have a familiar example of combined rolling and 
sliding in the mutual action of a pair of teeth in an ordinary 
spur-wheel; the motion of the balls in a bicycle bearing, 
again, is a case of combined rolling and spinning. 

The links of a certain class of mechanism are found to 
have such motions that their instantaneous axes all pass 
through a fixed point, while each portion of every link 
remains at its own constant distance from that point. Such 
motion is called spheric motion, because any given point 
on a link must be always on the surface of a sphere de- 
scribed about the fixed point as centre. It is evident that 



INTRODUCTORY CONSIDERATIONS. 19 

the most general case of spheric motion is that of a rigid 
body of which one point is fixed, and any kind of spheric 
motion can be made up by combining spins about axes 
passing through the fixed point. Plane motion may be 
looked upon as a particular case of spheric motion, in which 
the radius of the spheres is infinitely large. 

7. Freedom and Constraint. — We have seen that the 
essential feature of a kinematic pair is the mutual con- 
straint due to the forms of the two elements of which the 
pair is composed. Before considering the ways in which 
constraint or closure is actually applied it will be well to 
examine briefly the conditions on which the freedom of 
movement of a rigid body depends. 

The most general motion of a free rigid body may be 
looked upon as being a combination of three independent 
rotations about three rectangular axes, with three inde- 
pendent motions of translation along those axes. Such a 
body may then be said to have six degrees of freedom, one 
of which is taken away (or one degree of constraint is im- 
posed) when any one of these six modes of movement is 
rendered impossible. Suppose that the free rigid body is 
forced to touch a smooth fixed surface at one point, one 
degree of freedom is lost, for no translation can take place 
in a direction normal to the tangent plane to the surface at 
the points of contact. The three motions of rotation, how- 
ever, still remain possible, and so does motion of translation 
in any direction parallel to the tangent plane at the point 
of contact. A second point of restraint may be arranged 
so as to prevent one motion of rotation, or a second motion 
of translation, according to its position with regard to the 
first point of restraint and with regard to the form of the 
body. A third point of restraint causes the body to lose a 
third degree of freedom, and, finally, it will be found that 
all six degrees of freedom are lost, and the position of the 
body is fixed if six of its points are made to rest on six 



20 KINEMATICS OF MACHINES. 

portions of the surface of the smooth fixed body, and if 
these portions are properly formed and placed.* 

It may be shown that in general six conditions are 
required to completely determine the position of a rigid 
body, or, expressing the same thing in another way, six 
coordinates specify the position of one rigid body relatively 
to another, considered to be fixed. 

The definitions of a closed pair or of a closed chain 
given in §§ 2 and 3 thus mean that any element or link 
in a closed pair or chain may have only one degree of 
freedom as referred to the fixed element or link. 

Consider, for example, a screw turning in a fixed nut, 
like the screw of a micrometer gauge. The position of 
such a screw is determined exactly if an arm attached to 
its head is forced to remain in contact with a fixed stop on 
the body of the gauge, and we say, therefore, that such a 
screw has only one degree of freedom, inasmuch as its 
position is fixed by one point of constraint. The motion 
of a screw in its nut, a motion of translation accompanied 
by a definite and proportional motion of rotation whose 
axis is the direction of translation, is the most general kind 
of motion that can be possessed by a body having only one 
degree of freedom. 

The reader will notice that in two special cases, namely, 
when the pitch of the screw is infinite, and when the pitch 
is zero, the twisting motion of the nut becomes a mere 
translation or a mere rotation, both being specially impor- 
tant as plane motion involving one degree of freedom. 

In a similar way such a body as the connecting-rod 
of a direct-acting steam-engine is said to have constrained 
motion, having only one degree of freedom. The only 
possible motion at any instant for a given point on the 
rod is that of rotation about a certain virtual axis parallel 
to the axis of the crank-shaft. 

* See § 60, Chapter VII. 



INTRODUCTORY CONSIDERATIONS. 21 

Such a contrivance as a ball-and-socket joint cannot be 
regarded as a closed pair, for the ball has three degrees of 
freedom with regard to the socket. The ball has one point 
fixed, its centre, thus rendering all motion of translation 
impossible, and causing three degrees of constraint. The 
socket in fact might be replaced by three pairs of points 
touching the sphere at the ends of three diameters, each 
pair of points corresponding to one degree of constraint. 

Further examples may easily be imagined; the method 
of determining the conditions as to freedom and constraint 
in any particular case will be evident from the instances 
just given.* 

8. Elements and Pairs in Rigid Links. — It has been 
pointed out that the pairs of elements formed on the links 
of which a mechanism is made up are of two kinds, namely, 
lower pairs, in which the elements are in contact with 
each other over the whole or part of the area of certain 
surfaces, and higher pairs, in which such contact occurs 
only at certain points or along lines of points. 

In those portions of machines which are rigid the ele- 
ments must have forms w^hich can be readily produced by 
the ordinary processes of the workshop. Accordingly we 
find that their shapes are such as can be formed either in the 
lathe or the milling-machine, or by one of the many machine 
tools in which the cutting-tool describes a straight line with 
reference to the work. The rigid elements forming the 
closed pairs in machines therefore have in general for their 
working surfaces either surfaces of revolution, plane sur- 
faces, or screw surfaces. 

From the definition of Lower Pairs it is also plain that 
the forms of their elements must be such as to fit one another 
not only in one position, but in any position they may take 
up during their relative motion. It is plain also that two 



* The reader may refer to Thomson & Tait, Natural Philosophy, Part I, Sec- 
tions 195-201; also Tait, Enc. Brit., art. Mechanics. 



22 



KINEMATICS OF MACHINES. 




Fig. 8. 



surfaces of revolution, the one full and the other hollow, will 

fulfil this condition, and if properly 
formed, so as to prevent any sliding 
along the axis of revolution, will con- 
stitute a closed lower pair in which 
either element can only have con- 
strained motion relatively to the other. 
Such pairs are called shortly turning 
pairs, and Fig. 8 represents two bodies, 
a and h, so shaped as to form such a 
turning pair. The body h is partly 
cut away, to show more clearly the outline of a. 

The same condition (of fitting each other in any position) 
obtains in the case of a screw of uniform pitch and its nut. 
The relative motion is also constrained, as has already been 
stated, and consists of a motion of rotation around the 
axis of the screw, combined in a constant ratio with a motion 
of translation along that axis. Such a pair of screw sur- 
faces forms a screw -pair. ^ 

In general a lower pair formed by two cylindrical or 
prismatic surfaces will have constrained relative motion, 
because it will only be possible to give one body a motion 
of translation along the generating lines of the prism or 
cylinder relatively to the other body. If, however, the 
forms are circular cylinders, which are, of course, surfaces 
of revolution, then indefinite turning also is possible, the 
motion ceases to be constrained, and the pair is no longer 
closed. A pair of cylindrical or prismatic surfaces for 
which sliding only is possible is called a sliding pair (see 

Fig. 9). 

On examination it will be found that pairs of conical 
and other forms of surfaces generated by straight lines do 
not fulfil the conditions of continuous fitting or contact 
during motion, unless they are at the same time surfaces 



* See Chapter XL 



INTRODUCTORY CONSIDERATIONS, 



23 



of revolution. Non -cylindrical ruled surfaces in machines 
therefore have usually to take part in higher pairing. 

The three classes of lower pairs just discussed are then 
the only ones found in the rigid portions of machines. 
Examples of each kind will present themselves on examin- 
ing a few simple machines, and the means of constraint 
should be noticed in each case. For instance, in a shaft- 




FlG. 9. 



journal, endways motion or sliding of the shaft in its bearing 
is prevented either by making the diameter of the journal 
smaller than that of the adjoining portions of the shaft, or 
by securing collars on either side of the bearing. 

All the forms of ruled surfaces mentioned above, and 
•occasionally plane surfaces, surfaces of revolution, or screw 
surfaces, are found as portions of higher pairs, as well as of 
lower pairs. 

A simple arrangement of higher pairing can frequently 
be used to give motion of a kind which could only be other- 
wise obtained by a complex chain of lower pairs. 

It is important to notice that higher pairs give relative 
motion of a much more complex kind than is attainable 
by the use of lower pairing. This fact is pointed out by 
Burmester,* and is expressed if we say that supposing a 



* Lehrbuch der Kinematik, §§ 114, 116. 



2 4 KINEMATICS OF MACHINES. 

and h are two elements of a closed pair, and if a point A 
in a describes the same curve on 6 as a point Bmh (originally 
coinciding with A) describes on a, then the pair is a lower 
pair. If, on the other hand, A describes on 6 a line or 
curve different from that described by 5 on a, we have a 
case of higher pairing. Thus in the case of a lower pair 
no alteration of the relative motion occurs whether We 
consider one or the other of the elements as being the fixed 
one. 

Lower pairing is the more important from a constructive 
point of view, because the elements of a lower pair have a 
simpler relative motion, they are able to resist wear when 
transmitting heavy loads, and they can easily be made tight 
under fluid pressure. These are properties not possessed by 
higher pairs. 

9. Pairing of Non-rigid Links. — Passing on to the pair- 
ing of non-rigid links in mechanisms, it is found that these 
links may be classed under the following heads : 

(i) Flexible bodies, such as ropes, belts, or chains. 
These are almost invariably paired with cylindrical sur- 
faces on to or from which they unwrap or wrap themselves. 
Such pairing may be called tension pairing, since the rope, 
belt, or chain is necessarily in tension. 

(2) Pressure links, which continually exert pressure on 
the elements with which they pair. These links general^ 
consist of portions of fluid, such as air, steam, or water, 
and pair with the interior of the vessels containing them. 
Such a, pair is known as a pressure pair. 

Springs often form most important portions of mechan- 
isms and machines. They may be arranged so as to be 
in tension or in compression, and the resulting pairs may 
be said to be tension or pressure pairs, as the case may be. 

Actually all machine parts are elastic and so act to a 
certain minute extent as springs, but in kinematics we 
neglect all small changes of form, and consider such pieces- 
as being rigid, classing under the head of springs only those 



INTRODUCTORY CONSIDERATIONS. 25 

portions of machines whose elastic deformations tinder load 
are considerable in extent when compared with the proper 
motions of the other machine parts or links with which 
they pair. 

Non-rigid links will be considered at greater length sub- 
sequently. 

10. Classification of Mechanisms. — In attempting to 
classify mechanisms, which are made up of various kinds 
of links and involve so many kinds of pairing, we are im- 
pressed with the magnitude and complexity of the task. 
It may be said, in fact, that up to the present no wholly 
satisfactory kind of machine classification has been pro- 
posed. Some account of what has been done in this direction 
will be found in Chapter XIII ; for present purposes it will 
be sufficient to consider mechanisms under three heads. 

(i) Those involving only plane motion. These may be 
called shortly Plane Mechanisms, and form by far the most 
mportant and numerous class. 

(2) Mechanisms involving spheric motion, or, more 
briefly. Spheric Mechanisms. 

(3) Chains the relative motion of whose links is neither 
plane nor spheric, but of greater complexit3^ 

It is, however, to be understood that a mechanism of 
the third kind may contain certain links whose motion is 
plane or spheric, while any of them may include examples 
of both lower and higher pairing. 

A well-known instance of a spheric mechanism is Hooke's 
joint, the characteristic property of such chains being that 
the axes of the turning pairs they contain meet in a point. 
In the third class the most common examples are screw 
mechanisms. 

There is another method of classifying machines accord- 
ing to their geometrical properties, and according to the 
methods necessary for determining the various virtual 
centres of their links. Following this system, we should 
say that mechanisms of the Ftrst Order are those in which, 



26 KINEMATICS OF MACHINES. 

having given the relative position of any two links, the posi- 
tions of all the other links may be found by geometrical 
construction of straight lines and circles. From this it 
follows that in such mechanisms, having given the whole 
mechanism in one position, we can find geometrically all 
its other possible positions, and the virtual centre of each 
link relatively to every other. Mechanisms not possessing 
these properties belong to higher orders, and are of com- 
paratively infrequent occurrence. 



CHAPTER II. 
POSITION, VELOCITY, AND ACCELERATION. 

II. Velocity. — While Kinematics in its general sens^ 
comprises all kinds of problems dealing with pure motion, 
the niimber of such problems falling within the province of 
the Kinematics of Machines is somewhat limited. We 
shall consider in this chapter some elementary notions con- 
cerning velocity which are applicable to the purposes of 
the Kinematics of Machines. Methods of studying the 
position and motion of a point or rigid body from a geomet- 
rical point of view have already been indicated; it now 
remains to investigate not only the amount by which such 
position is changed during motion, but the rate of such 
•change of position. Going a step farther still, it may be 
asked, does such velocity increase, diminish, or change in 
any way as time goes on, and if so, at what rate? 

The rate of change of position of a point or body is 
called its velocity. A body, as we have seen, may change 
its position by a motion of translation, or by one of rotation. 
Hence we distinguish between linear and angular velocity. 
The former is measured by the space passed over in unit of 
time, and is usually expressed in feet per second, although 
other units, such as miles per hour or knots, are adopted in 
special cases. The latter is measured by the angle de- 
scribed in unit of time, the natural unit being therefore one 
radian per second. Engineers, however, commonly measure 
angular velocity in revolutions per minute. Either kind 
of velocity may be uniform or variable. 

It is important to note that the term velocity involves 

27 



28 KINEMATICS OF MACHINES. 

the ideas of both speed, direction, and sense. In other 
words, a velocity is a vector quantity, and, Hke other vector 
quantities, may be represented by a straight hne of definite 
length, this length being proportional to the speed, or mag- 
nitude of the velocity, measured in feet per second, radians 
per second, or whatever units are to be employed. 

In the case of linear velocity the direction of the vector 
or straight line representing the velocity on the diagram is 
taken to represent the direction of the motion. Thus, for 
example, we might draw upon a map a line running east 
and west, and 2 inches in length, and take this line as repre- 
senting a linear velocity of 2 miles per hour, or 2 feet per 
second, either from east to west, or from west to east. The 
sense of the motion raay be either from east to west, or from 
west to east. In order to indicate the sense, we place upon 
the line a small arrow-head so as to show the point towards 
which the body is moving (see Fig. 12). 

In the case of angular velocity the direction of the vector 
on the diagram would be taken to represent the direction 
in space of the axis about which the spin or rotation is taking 
place, and a line similar to that mentioned above would 
mean a spin of two radians per second, or two revolutions 
per minute, according to the scale, about an axis lying east 
and west. This rotation may be either right-handed or 
left-handed, and it is therefore customary to indicate the 
sense by placing the arrow-head in such a fashion that the 
spin will appear to be right-handed, or clockwise, when 
looking along the axis and following the arrow-head. 

It is plain that in this manner a velocity, whether linear 
or angular, may be completely represented by a vector^ 
having magnitude, direction, and sense. 

12. Uniform Velocity. — ^A body having uniform velocity 
(whether angular or linear) performs equal changes of posi- 
tion in equal times. If the body has a uniform linear veloc- 
ity V, it describes a distance vt in time t, where i is any num- 
ber of units of time. Calling 5 the space described, we have 



POSITION, VELOCITY, AND ACCELERATION. 29 

therefore 5 =vt. Similarly, if the uniform velocity is angular 
and is denoted by w, any line on the body in a plane per- 
pendicular to the axis of rotation describes w radians in 
each second and therefore ojt radians in t seconds. Hence, 
calling d the angle described in t seconds, we have 

d = wt 

If a point, at distance r from the centre about which it 
moves in a circular path, has a linear velocity v, its angular 
velocity is measured by the angle subtended at the centre 
by the path it describes in one second. Hence 

V 

0)=— or v = (ji}r, 
r 

13. Variable Velocity. — In general a moving body 
varies its speed as well as its direction of motion. It is easy 
by observing the time taken to travel over a known dis- 
tance, for example in a train, to calculate the average speed 
of the train during the interval considered. This does not 
tell us, however, the actual speed of the train at any instant 
during the interval of time, which may be quite different 
from the average speed. 

The velocity at any instant, or instantaneous velocity, is 
measured by the space (or angle, as the case may be) which 
would have been described in a unit of time if the motion 
had continued uniformly, during that interval, at the same 
rate as at the instant considered. The word instant is here 
used to mean an indefinitely small interval of time. 

We are not able to measure the distance (or angle) de- 
scribed during an indefinitely small interval of time, and 
therefore have to obtain the value of the instantaneous 
velocity of a body in another manner. 

This will be best understood by a numerical example. 
Suppose that a man in a street-car at 12 o'clock finds that 



30 



KINEMATICS OF MACHINES. 



in lo seconds the car traverses a distance of 200 feet. This 
gives 20 feet per second as the average speed during the 10 
seconds after 12 o'clock. Suppose that other observations 
taken during the first i^, 2, and 4 seconds showed that 
during these times the car travelled 30, 48, and 100 feet, cor- 
responding to average speeds of 25, 24, and 21.75 ^^^^ P^^ 
second. It is evident that the speed must really have been 
continually diminishing, and that the shorter the time dur- 
ing which the observation was made, the more nearly do we 



EEET PER 
SECOND 



■f 



_l-. 



30 






















. 










'^'^ 


»^^ 


















-^ Ci 


mvE . 


»_OF AVERAr 


E SPEED 






on 










■ 






k 


liy) 














H 


r--A" 

























£ CO 




a 














CE Q 




z CO 











3 5 




5 Q 






z 




Q 




-3 z 






2 2 








Q 
LU (0 






= 5 
So 


10 






UJ 






UJ UJ 
U 00. 


^^ 




•3m\ 






»2 






UJ * 








2S 




1- 






UJ . 

UJ u. 










< 






> 


















< 






i 


, 




^ 


, 






\ 





12 H. M, SEC. 



10 TIME 
SECONDS 



Fig. 10. 



obtain the speed at which the car must have been travelling 
at 12 o'clock. To arrive at this more exactly, since we 
cannot measure the distance passed over in an infinitely 
smah interval of time, we plot a curve from our observations, 
as in Fig. 10, and see that the speed at 12 o'clock must have 
been 27 feet per second. In mathematical language, if ^s 
be the distance traversed in a small interval of time Jt, the 

Js 
average velocity during that small interval is — -, while the 

velocity at the instant beginning the interval is measured by 



POSITION, VELOCITY, AND ACCELERATION. 31 

diminishing At indefinitely, and finding the limiting value 

A /7c 

of — , or, in the language of the calculus -j- . Thus 
At at 



ds 
v=—. 
dt 



The same reasoning, of course, applies m. the case of 
angular velocity, where we should write 



dd 
iO = — . 
dt 



Compare these with the corresponding expressions in the 
case of uniform velocity. 

14. Uniform Acceleration. — A body moving with uniform 
acceleration changes its velocity by equal amounts in equal 
times. Thus suppose that in time t the velocity changes 
from V, to v^ ; we have, if a is the acceleration. 






Agam, the average velocity during the time / is — i, the 

2 

arithmetical mean between the mitial and final velocities; 

hence if 5 be the space described, 



5=^-i±^'/ (2) 



From these two expressions we find 



s=<^^ (3) 

2 a 



32 KINEMATICS OF MACHINES. 

But v^ = v^-\- at. Substituting in (3) , we get 

s=v,t-\-^e (4) 

In the case of angular velocity precisely similar relations 
hold, so that, calling oc the uniform angular acceleration, co 
the angular velocity at the beginning of the time t, and 
the angle described, we have, instead of (4), 

d = w^t + -t' (4a) 

To express the velocity in terms of distance (or angle) 
and initial velocity we shall have instead of (3) 

^2' = ^i'+2«<9, ...... (3a) 

while the expression connecting velocity, acceleration, and 
time is 

(o^ = oj^-{- at. (la) 

As an example of the use of these expressions, suppose a 
wheel is revolving thirty times per second and comes to rest 
in T 2 seconds. How many revolutions will it make in coming 
to rest if uniformly retarded? 

We have aj^ = iJL)^ + at\ hence 

12a-\- 30 X 271 = 0, 

and a = — — - = — 15.71 radians per second per second. 

Again, oj^"^ = uj^^-^ 2 ad; hence 

(6or)^— lo.T^ =0, 

and 0=- — ^- =360- radians. 

lOTT 

Hence the wheel comes to rest in 180 revolutions. 



POSITION, VELOCITY, AND ACCELERATION. 



33 



Again, a train starting from rest has a imifomi accelera- 
tion of half a mile per hour per second. How far will it 
have travelled before attaining a speed of 30 miles per hour, 
and in what time will this occur ? 

In the equation (4) above we have s^v^t + ^af^. 

2640 



Here v^=o, t evidently will be 60 seconds, and a 
0.733 ^^^^ P^^ second per second. Thus 



3600 



^^0.733X3600^^^^8 feet. 

2 

It should be noted that it is as incorrect to speak of 
an acceleration of so many feet per second as it would be to 
say that a body has a velocity of so many feet, without men- 
tioning the unit of time. 

15. Acceleration in General. — The determination of 
velocity and acceleration in the case of non-uniform or iion- 




0S 



Fig. II. 



uniformly accelerated motion will be discussed later. 
Acceleration is defined generally as rate of change of velocity 
with regard to time, and so far we have used the term as 
meaning change in the magnitude of the velocity, whether 



34 KINEMATICS OF MACHINES. 

linear or angular. Strictly speaking, however, a change 
in the direction of motion in linear velocity, or in the position 
of the axis of rotation in angular velocity, is also an accelera- 
tion. In fact, a point travelling in a circular path around 
a fixed point has an acceleration impressed upon it, although 
its angular velocity may be uniform, and such acceleration 
is called radial, for reasons which will presently be seen. 

In Fig. II let AB represent a portion of the curved path 
alng which a point is travelling with a linear velocity v, 
whose direction is continually changing. Let p be the 
radius of curvature OP of a very small portion PQ, and 
the centre of curvature, cp being the very small angle be- 
tween the tangents at P and Q, an angle so small that the 
arc PQ is not sensibly different from its chord. 

Consider the acceleration in a direction parallel to PO. 

The time taken for the particle to travel from P to Q will be 

PO . . 

— —. But during this time the distance traversed under ac- 

V 

celeration a is P^Q parallel to PO. Hence (if a is constant) 

It is known that for very small angles the numerical 
value of the sine of an angle is sensibly the same as the angle 
itself (of course expressed in circular measure) . Also in the 
figure if we make (p small enough, MQ^^PQ, the error in 
this statement diminishing as <p diminishes. Hence for an 
indefinitely small value of (p we may say that 



2P1Q P,Q ,^ I ^Q,, I I a\. ^' 

TQ^-PQ^^'^''PQ^0P''PQ^'P"^^^^'''''^7^ 

The statement is exactly correct, and a is the radial 
acceleration at P, because we have taken ^ as being the 
angle described during an indefinitely small interval of 
time. 

The earth's equatorial radius is 4000 miles, and the 



POSITION, VELOCITY, AND ACCELERATION. 



35 



earth makes one rotation on its axis in about 86,200 seconds. 
What is the radial acceleration of a particle on the earth's 
surface at the equator? 

Linear velocity of a point at equator 

27rX4oooX S280 J. ^ 1 

= -J— reet per second. 

86200 

Thus ^'_ (^^)'X4oooX528o 
p 86200^ 

= 0.112 feet per second per second. 

It is often necessary to find the acceleration of a body 
along its virtual radius; this is of course determined in 
exactly the same way as the radial acceleration with regard 
to a permanent centre. 

16. Composition of Velocities and Accelerations. — It has 
been already pointed out that velocities, whether linear or 
angular, can be represented by straight lines of definite 
length, sense, and direction, and are in fact vector quantities, 
as distinguished from scalar quantities, such as mass, 
energy, and so on which have simply numerical values. 
Accelerations are also vector quantities. 

The resolved part of a vector in any new direction is 
found by projecting its original length on the new direction. 



w- 




FiG. 12. 



If, for example, a ship is proceeding northeast at a speed of 
13 knots, represented by the vector AB (i knot being a 



36 



KINEMATICS OF MACHINES. 



speed of 6080 feet, or i nautical mile, per hour), its resolved 
velocity in a northerly direction is represented by 

AC =AB cos 45° = 13 X0.707 =9.19 knots. 

This shows that each hour the position of the ship is 9.19 
nautical miles farther to the northward. 

Again, suppose that the ship, still steering N.E. at the 
same speed, runs into a current whose speed is 4 knots due 
east, what will be the real velocity of the ship relatively to 
the earth? Relatively to the water its speed is still 13 knots 



w- 




FiG. 13. 



in a N.E. direction, but the water is itself moving, and at 
the end of the hour the ship will evidently be at D, a position 
obtained by measuring 4 nautical miles east from B . On cal- 
culation it will be found that at any time during the hour the 
ship has been moving relatively to the earth along the line 
AD, and its real speed over the ground (about 16 knots) will 
be measured by the length of AD, the third side of a triangle, 
whose other two sides represent respectively the velocity 
of the ship relatively to the water, and the velocity of the 
water relatively to the earth. We say, then, that the vector 
AD represents the resultant of the two vectors AB and BD, 
obtained by the process of vector addition. 




POSITION, VELOCITY, AND ACCELERATION. 37 

The above example deals with plane motion in a straight 
line only. But if we are treating of the motion in space of 
a body having six degrees of freedom, its motion may be 
considered as made up of three motions of simple translation 
and three motions of rotation, which, when compounded 
according to the method just explained, constitute the 
actual motion of the body. 

It must not be forgotten that the resultant of two or 
more angular velocities can be found in exactly the same 
way as for linear velocities. As already explained, it is 
customary to indicate an angular q 
velocity by a vector (as in Fig. 14), 
representing the numerical value of 
the velocity by the length AB, the 
direction of the axis by the direction 
oi AB, and the sense of rotation by ^^ 

drawing AB in such a manner that fig. 14. 

the rotation is clockwise, or right-handed, when looking from 
A to B. It is often necessary to compound or to resolve 
spins or angular velocities according to the method of 
vector addition, which will be already familiar to most 
readers under the name of the triangle of velocities, or 
the parallelogram law for the composition of vectors. 

17. Resultant Acceleration. — In Fig. 15 let AB repre- 
sent the original velocity of a particle, and suppose that 
accelerations represented by BC, BD are impressed upon 
the particle. Then BC and BD may be taken to represent 
the velocities generated in one second, corresponding re- 
spectively to the two accelerations. 

If now the acceleration BC had alone acted on the point, 
its velocity at the end of one second would have been AC. 
Again, ii AC had been the original velocity and an accelera- 
tion BD had been impressed, the final velocity at the end 
of one second would have been AD\ where CD' is equal and 
parallel to BD. The two accelerations, therefore, have 
changed the original velocity from AB to AD\ But this 



38 KINEMATICS OF MACHINES. 

effect would have been produced by compounding with AB 
for one second a velocity BD\ and we may therefore look 
on BD' as representing the change of velocity in one second, 
due to the action of the accelerations BC and BD. In 



"if o 

B D 



Fig. 15. 



other words, BD' is the resultant of the accelerations BC 
and BD. 

The general rule for the composition or addition of vectors, 
then, is that the resultant of two vectors is the diagonal of a 
parallelogram (or the third side of a triangle) of which the 
two components form the two adjacent sides. In this way 
we can find the resultant of any number of velocities, or 
of accelerations, either linear or angular. The same rules 
apply to the composition of any other vector quantities. 

18. Diagrams of Displacement and Velocity. — In study- 
ing the motion of a body, whether linear or angular, it is 
necessary to know the position of the body at every instant 
during the motion, if we desire full information as to its 
velocity and acceleration. We have seen that if we only 
know the position of the body at certain times we can obtain 
the value of the average velocity between those times, but 
cannot tell exactly how the real velocity has changed. 

It would of course be very cumbersome to have to state 
in words or figures a sufficient number of particulars to give 
us a practically complete knowledge of the position, velocity, 
and acceleration of a body, and therefore in such cases 



POSITION, VELOCITY, AND ACCELERATION. 



39 



graphic methods of representation and calculation are gen- 
erally adopted. 

For example, in order to determine the velocity of a 
body whose changes of position are known, such a diagram 
as Fig. i6 is constructed. Two axes, OA, OB, are drawn at 
right angles, and distances measured parallel to OA accord- 
ing to any convenient scale are considered to represent time, 




Fig. i6. 

while lengths measured parallel to OB represent either the 
distance that has been traversed by the body, reckoning 
from some known position, or the angle turned through by 
the body at any given instant. This quantity we may call 
the displacement of the body, and it may be either linear or 
angular. 

For instance, from the figure we see that after the lapse 
of 3 seconds the body in question has moved lo feet from 
its original position, and we might give the information 
■contained in the diagram in a less complete form in the 
shape of a table, thus: 



Time o i 2 

Displacement. 005 



345 seconds. 
10 12 12 feet. 



From the diagram, however, we are enabled to gather 
further particulars, for it is plain that the curve of displace- 



40 



KINEMATICS OF MACHINES. 



ment during the second and third seconds is straight, i.e.^, 
distance is increasing proportionally to time, or the velocity 
is uniform and the speed 5 feet per second. During the 
fourth second the distance increases more and more slowly, 
and then remains constant, hence the velocity diminishes 
and finally ceases. Thus plainly, in order to find out the 
velocity at any instant from such a diagram, we have only 
to determine the rate at which distance (or angle) is increas- 
ing or diminishing at that instant. We shall see that this^ 
information can be obtained from the form of the curve. 




a b 



Fig. 17. 



AXIS OF TIME 



In Fig. 17 let ABC be any curve of displacement on a. 
time base, and let Aa, Bb represent the distances (or angles) 
corresponding to the times Oa, Ob. We wish to determine 
the velocity at the point A, i.e., after the lapse of the time Oa, 

It has already been pointed out that to find the instan- 
taneous value of a velocity, or the velocity at any instant, 
we must take what is really the average velocity during an 
infinitely small time, or, if Js be a small change in position 
and Ji the corresponding interval of time, the instantaneous 

. . . Js 

velocity is the limiting value of — . In the figure let A and 

B be very close together; evidently BD =Bb — Aa = Js, the 



POSITION, VELOCITY, AND ACCELERATION. 41 

change in distance during the time At. The ^mall interval 

of time At is represented by the length ab=AD. Hence 

the ratio 

BD _As 

AD~.At' 

Draw the straight line AB and produce it to cut the axis of 

time in E, making an angle AEa = (p. Then -— =-r^ = t^ 

At AD Ea 

= tan <p. Now suppose we diminish J ^, making 5 approach 

A more and more nearly, as is shown on a larger scale in 




Fig. 18. 
Fig. 18. The chord AB becomes AB^ and, as J^ diminishes, 
approaches more and more closely to the tangent to the 
curve at A, and, if At is made infinitely small, AB will 

As 

comcide with that tangent. Still, however, the value of — 

is shown by the numerical value of tan <p, and in the limit 

velocity = tan ^ = -- •* 

Accordingly we may say that to find the velocity of a 
body at any instant from its diagram of displacement drawn 
on a time base, we have only to draw at the point correspond- 
ing to that instant a tangent to the curve. The slope of 
that tangent, as measured by the tangent of the angle it 
makes with the axis of time, is proportional to the velocity, 
and indeed represents the velocity numerically if the dis- 
tances are measured to their proper scales. For example, 
at C in Fig. 16, to the scales marked, tan (p has the value 

I o feet 

and shows, therefore, that at 3 seconds the body in. 

2 sec. 

* See § 13. 



42 KINEMATICS OF MACHINES. 

question had a velocity of 5 feet per second. If a scale of 
miles had been marked along the axis of distance, while 
hours had been measured along the axis of time, our velocity 
would have been read in miles per hour. 

The reader should notice that at the point F in Fig. 1 7 
the displacement has ceased to increase with increase of 
time, and, is about to decrease ; the body has in fact reached 
its maximum distance from its starting-point. At this 
point the body must of course cease to be moving for an 
instant, which is shown by the fact that the tangent to the 
curve at F is horizontal, corresponding to zero velocity. 
After F the velocity will of course have to be reckoned 
negative, since distance is now diminishing as time goes on. 

If the velocity at every instant could be measured in this 
way and a new curve drawn on a time base, having ordinates 
at each instant proportional in length to the velocity at that 
instant, we should obtain a curve or diagram of velocity. 
Actually we obtain only a sufficient number of values to 
give us a series of points on the curve, through which the 
curve can be drawn. This has been done as an example 
in Fig. 19. The full curve is a diagram showing the dis- 
tance from London at times between 11.50 a.m. and 12.20 
P.M., on a certain date, of the London and Exeter express 
on the Great Western Railway, the times of passing various 
stations and mile-posts having been carefully noted. From 
this full curve has been drawn a dotted one, the height of 
which at any point is proportional to the speed of the train 
at that time ; this dotted line in fact is a curve of velocity. 

It will be seen that the curve of displacement slopes 
continually upward, showing that the train did not stop 
during the time considered. Its speed, however, was very 
variable. The train performed the whole journey from 
London to Exeter, 194 miles, in 3 hours 38 minutes without 
a stop ; thus the average speed was 53.4 miles per hour. At 
about 12.14, however, the speed for a very short time is seen 
to have exceeded 80 miles per hour, as shown at D, while 



POSITION, yELOCITY, AND ACCELERA TION. 



43 



o 





S § 8 fe S S 2 
1 1 1 1 III 


00 ;dcd 
? 3? 




SCALE OF SPEED 


IS 




•-l M. 


5^ SS 


^ H-L 


00 oo o to O O 


o •-' 


?•?? 


O W O Ot O or 


i <=> 


2 Of" 




r— 


• O 


. SCALE OF DISTANCE 


o s 


m 


<p\\ \ 


Z 1- 

2 m 
O CO 

z 


•-> 


■-«N\ . X 




Ff 


'•s-v^ * 




cJr" 


\v \ 




Of 


\ * 

\ ^ 

\ \ 

\ \ 




P 






w^ ^w 












\ ^"^ 




is 




< 


\ ''" 






V ^ 






V^ \ 






\<f- } 






\ 1 






\o / 






V^ / 






\ / 




8" 


\ yf 










V-^ /'^ 






\ 4^° 






\ /v^ 






\ / 






\ / 




??- 


V 




^i 


-' ^\ 




o 


\ N 










\ 


.->0 


fe 


-\ 




fe- 


X \ 

X \ 
• I 




T-fe 




^i" 






H 


""/"""^--^ \ 




^ 


^"^^^ \ 





44 KINEMATICS OF MACHINES, 

shortly afterwards, at about 12.19, a signal-check caused the 
speed to be reduced to about 10 miles per hour, as shown 
at C. The method of finding points on the curve of velocity- 
is shown at B. The ordinate at B is proportional to the 
ratio A a : Ea, where A£ is a tangent to the displacement 
curve at A. 

By the construction of such curves we can trace out the 
whole history of the motion, and if a sufficient number of 
points have been taken, and our drawing has been accurate, 
the results will be trustworthy for all practical purposes. 

It is often difficult to draw the tangents to the curve 
correctly enough, especially if the slope of the curve is small, 
and it is usual to adopt another construction, to be explained 
later, which avoids the necessity of drawing the tangents by 
guesswork. 

It is to be noted that diagrams of displacement may 
quite well be drawn, in which ordinates represent angles 
instead of distances, and from such diagrams angular 
velocities can be obtained exactly as described above. 

19. Diagrams of Acceleration. — If a diagram of velocity 
on a time base be drawn, the curve of acceleration can be 
obtained from it by an application of the same method 
adopted for getting the velocity curve from that of displace- 
ment. In Fig. 20 let OAB be a curve of velocity. At any 
point A the rate of change of velocity is the limiting value 
of the ratio of the small change Av in velocity to the small 
interval of time At in which such small change occurs, and 
by similar reasoning to that in § 18 it will be seen that 
this ratio is numerically equal to the tangent of the angle 
of slope of the curve at the point considered. In the figure, 
then, the acceleration at A is represented by tan a. Plainly 
at such a point as C, where the velocity has a maximum 
value (having ceased to increase and being about to de- 
crease), there will be no acceleration, and the acceleration 
curve will pass through the time axis, as at c. Further, it 
will be noticed that where the velocity is increasing uni- 



POSITION, VELOCITY, AND ACCELERATION. 



45 



f ormly (as between D and E) , and the curve of velocity is 
therefore straight, the acceleration curve becomes a horizon- 
tal straight line, as at de. 




Fig. 20. 



In practice, when such diagrams have to be drawn 
either for curves of velocity or of acceleration, a somew^hat 
different method is adopted (as shown in Fig. 21), based on 



46 



KINEMATICS OF MACHINES. 




20 SEC. 



20 SEC. 



Distances from 

Rest. 
5 sec. 66 ft. 
lo 245 

15 490 

20 750 



Velocity. 
5 sec. 19 m. p.h. 
10 30 

15 36 

20 39 



Acceleration. 
5 sec. 3 m. p. h. per. 
10 1.4 [sec. 

15 .9 

20 .6 



20 SEC. 



Fig. 21. 



POSITION, VELOCITY, AND ACCELERATION. 47 

the same principle, but avoiding the necessity of drawing 
a number of tangents to the curve, many of which can only 
be determined approximately. 

The diagram shows a curve of displacement* for an 
electric street-car starting from rest. The data were ob- 
tained from tests of a special car designed for an initial 
acceleration of 3 miles per hour per second. 

To construct from such a curve the corresponding curve 
of velocity, the time shown by the diagram is divided into 
a number of small intervals, in this case of 2 seconds each, 
as at, a6. On measurement the length he is found to repre- 
sent 100 feet and is the difference between aa' , or hU , and ch'. 
Now aa' is the distance moved by the car during the first 
14 seconds. Thus ch represents the distance traversed during 
the fifteenth and sixteenth seconds, and accordingly the 
average veloeity for those tw^o seconds was ^f^, or 50 feet per 
second. 

In the figure an ordinate ef of length five times he has 
been marked off at the point corresponding to 15 seconds 
from the start, and its extremity gives one point on our 
curve of velocity. In the same way jk has been made 5 X g/j, 
and so on. 

A curve drawn through the points thus found shows 
approximately what was the velocity at any time after the 
start. We say approximately, because the aetual veloeity 
Sit the middle of a 2 -second interval would only be equal to 
the average veloeity during such an interval if the straight 
line joining ac (for instance) had been parallel to the tangent 
to the displacement curve at d; that is, if the points a", /, 6" 
had been in a straight line. We know that actually this 
may or may not be the case, but by taking sufficiently small 
intervals of time we can reduce the error from this cause to 
any desired extent, until in fact it becomes neghgible. 

In order to obtain from the velocity curve that of accel- 

* Taken from Engineering News, Oct. 14, 1897. 



48 KINEMATICS OF MACHINES. 

eration, an exactly similar procedure is employed. The 
length Im, for example, has been set up and exaggerated 
fivefold at np, and is proportional to the change of velocity 
during the tenth and eleventh seconds. 

Having drawn our curves, it becomes necessary to deter- 
mine their scales, that of the original diagram of displace- 
ment being known. 

The displacement diagram was drawn originally * on such 
a scale that he (representing loo feet) was actually half an 
inch; the scale of distance was then 200 feet to the inch. 
The scale of time was i'' = 5 seconds, or i second = ^ of an 
inch. A length he transferred to the velocity diagram then 
represented (if ah = 2 seconds) a velocity of 50 feet per 
second, giving a scale of i'' = 50 feet per second, or 100 feet 
per second to the inch. For clearness this was exaggerated 
in the figure, so that 20 feet per second = 1 inch. In the 
same way the scale of the acceleration curve was made 
such that five inches = 10 feet per second per second. 
Scales of miles per hour and miles per hour per second have 
also been marked for comparison. 

In general, then, the scales of such diagrams may be de- 
termined as follows : 

Let the displacement diagram be drawn to a distance 
scale of I inch=/ feet, and suppose the short intervals of 
time during which the average velocities are estimated are 
each n seconds. 

An ordinate of i inch in length on the displacement 

diagram when transferred to the veJocity diagram then 

represents a velocity of / feet in n seconds, i.e., l/n feet per 

second. The scale of the velocity diagram is then n/l inches 

= I foot per second. 

Thus in the figure above we should have, if the ordinates 
were not exaggerated, a velocity scale of -y^tt inch = i foot 
per second, for / = 200 and n = 2. It was drawn actually to 
a scale of ^V inch = i foot per second. 

* It is of course reproduced here to a smaller scale. 



POSITION, VELOCITY, AND ACCELERATION. 49 

Considering next the scale of the acceleration diagram, 
suppose that on the velocity diagram the velocity scale is 
I inch =m feet per second, while the small intervals of time 
are as before n seconds each ; then an ordinate of i inch on 
the acceleration diagram will represent an acceleration of 
m/n feet per second per second. 

In the figure w = 20, n = 2, so that the acceleration scale 
would naturally have been |-o = lo feet per second per second 
to the inch, had it not been exaggerated for clearness to 2 
feet per second per second to the inch. 

20. Diagrams on a Displacement Base. — Diagrams of ve- 
locity and acceleration may also be drawn on a displacement 
base, in which case lengths measured horizontally are pro- 
portional to distance traversed or angle described ; such a 
construction is frequently very useful. 

We have seen that on a time base the velocity curve for 
a body having uniform acceleration will be a straight line, 
passing through the origin of the two axes if the body has 
no velocity when time is reckoned zero. Velocity is then 
proportional to time. Suppose, however, that we consider 
the way in which velocity varies with regard to displace- 
ment in such a case. 

Let a be the constant acceleration, v the velocity attained 
from rest after moving for a time t; then by our definition 

a = -. Now the distance moved by the body will be numer- 
ically equal to the average velocity ( - J multiplied by the 

-H, ^'^ 25 

time ; thus 5 = — or z; = — . 
2 t 

But v = at\ hence 

, 2ast 
v^ = =2as. 



Therefore, the acceleration being constant, the displace- 
ment varies as the square of the velocity. In the figure the 



50 



KINEMATICS OF MACHINES. 



acceleration is 1.5 feet per second per second, and the dis- 
tance traversed in the first two seconds (during which a 



FEET PER 
SECOND 



> 
»- 

8 4 

UJ 

> 



12 3 4 5 6 



FtET'PER 
SECOND 



7 SECONDS 



TIME 



6 
5 

t 4 

§ 

UJ 8 

2 
1 
































































^ 












y 
















/ 


/ 
















( 



















3 4 5 6 

DISPLACEMENT 

Fig. 22. 



9 FEET 



velocity of 3 feet per second has been attained) is |X2 = 
3 feet. In 3 seconds, the velocity being 4.5 feet per second, 

a distance of-^X3=6.75 feet will have been described, and 



POSITION, VELOCITY, AND ACCELERATION. 



51 



in I second, the velocity being 1.5 feet per second, a distance 

of -^X I =0.75 feet will have, been covered. The diagram 
2 

of velocity and displacement shown in the lower part of 
Fig. 22 will be found to express these relations, and 
the velocity curve on a displacement base is not a straight 
line, but a curve whose ordinates are proportional to the 
square roots of the abscissae. 

21. Acceleration Curves on a Displacement Base. — Let a, c, 
(Fig. 23), be two points on a velocity curve drawn on a 
displacement base. The difference of the ordinates he repre- 
sents the change of velocity. Draw the straight line ac, bisect 




Fig. 23. 

it by the line df at right angles, and draw de perpendicular 

to the axis of displacement. Then -~ =^r, since the triangles 

ed ab 

abc, def are similar. 

Assuming that the velocity changes uniformly, while 
the moving body describes the distance represented by 
a'6', let 5 be the space described in time t, while the velocity 
changes from v^^ to v^. Then we know that 

s=-^ — H and t= — ; — . 
2 ^2 + ^1 



52 KINEMATICS OF MACHINES. 

Again, the acceleration 



a- 


J^l 






_(^2- 


-^1 


(^2 + 


^l) 



hence 

a 

2S 

But in the figure aa'=v^, cb^ =v^, so that V2 — v^= be, 
-^ ^ = J^, and s = ab; thus 

2 

. ,. bcXde r 

acceleration = — = ef. 

ab 

Actually, of course, the velocity in general does not 
change uniformly; we may, however, take a indefinitely 
close to c, so that the straight line ac becomes a tangent to 
the velocity curve, and df becomes the normal to that curve, 
while ef is the subnormal at d and represents the accelera- 
tion. 

We find, then, that in the case of a velocity curve drawn 
on a displacement base the subnormal at any point repre- 
sents the acceleration. It only remains to determine the 
scale on which, for example, ef represents the acceleration 
at ^. 

Let the scale for velocity be i inch = m feet per second, 

while the distance scale is one inch=n feet. Let ab, be, de, 

and ef be measured in inches. Then, numerteally, v^~v^ = 

v -\-v 
be X w feet per second , — ^ =deXfn feet per second, and 

2 

s = abXn feet. But we have just seen that 

acceleration = — ^^— -^ — 

SX2 

beXmXdeXm 



abXfi 
= efX 



m^ 



n 
ef being also measured in inches. 



POSITION, yELOCITY, AND ACCELERATION. 53 

In Fig. 24 is given the curve of velocity for a cable -car 
from starting to stopping on a run of 200 feet, and from it 
the curve of acceleration has been drawn, the construction 
for one ordinate being shown. Note the high positive accel- 
eration at the start, indicating a considerable jerk, followed 
(after the first 35 feet) by a small and variable value of the 
acceleration, sometimes positive, sometimes negative. The 
stoppage of the car is accomplished in the last 35 feet, 
and is shown by the negative acceleration or retardation 
during that portion of its travel. 

As originally drawn, the velocity scale was i inch = 5 
miles per hour = 7.33 feet per second, while the distance scale 
was one inch = 50 feet^ Accordingly the acceleration scale 

was I inch = ^^^^ — ^^ = 1.075 feet per second per second, 

or o. 73 2 mile per hour per second. The greatest acceleration 
is then about 3.3 miles per hour per second, and the greatest 
retardation about 2.2 miles per hour per second. 

It should be noted that the curve of velocity on a displace- 
ment base, when acceleration is constant, is a parabola, this 
being the curve one of whose characteristic properties is 
the constancy of the subnormal.* 

22. Polar Diagrams of Displacement, Velocity, and Ac- 
celeration. — Besides employing the methods just given for 
drawing curves of displacement, etc., it is often useful, 
especially in considering periodic motion (in which the same 
circumstances or conditions as to velocity or displacement 
repeat themselves at regular intervals of time), to draw 
diagrams in which a radius vector represents displacement, 
velocity, or acceleration, while the angle turned through by 
such a radius is proportional to time. 

In Fig. 25 suppose that a line ON turns about the point 
O. starting from the initial position OM ; the angle 6 it has 
described, when it has reached any position such as OA , being 
proportional to an interval of time t^, during which a body 

* See Fig. 22. 



54 



KINEMATICS OF MACHINES, 




Fig. 24. 



POSITION, VELOCITY, AND ACCELERATION. 55 

whose motion we are considering has moved to a distance s^ 
from its starting-point. Mark off a distance OA =s^. Then 
a curve, such as NAB, drawn through successive positions 







Fig. 25. 

of ^, is a polar diagram of displacement for the body con- 
sidered. Let OB = ^2, while the angle BOM = t^. 

Then the average velocity between the times t^ and t^ will 

be represented by - — z — ==^7;-, = -^ ^ , Making OC = OA , 

angle BOJi ^2~^i 

we have, therefore, 

BC As 



average velocity = 



angle BOA Ad 



Join AC, and draw the straight line BA, Now \i Ad \s 
a very small angle, the length of the straight line AC does not 
differ sensibly from that of the arc of a circle of radius OA, 
so that we may sav that if i^ is small, 

Ad=—' 
AO' 

As _ BC.AO 

Ad~ AC ' 



56 KINEMATICS OF MACHINES. 

Bisect AB in D, and draw OD, DE, OE, respectively 
perpendicular to AC, AB, BC, so that the triangles BCA, 

EOD are similar; then - ^ =^^^- and 

AC DO 

^' _0E 

Next suppose that the interval of time represented by 
Ad becomes indefinitely small, so that the points B and A 
coincide, and AB becomes a tangent to the curve. J^and 
Ad will both become infinitesimal in magnitude, but their 
ratio (which now represents the velocity the body has when 
a time represented by has elapsed) is finite, and its value 

is measured by the limiting value of — =— — when A0= DO. 




Fig. 26. 

This is of course OE. Hence the velocity is measured by 
the length of OE when taken to the proper scale and drawn 
perpendicular to OA, and to find the velocity correspond- 
ing to any point A on a polar displacement diagram, we 
draw a normal to the curve, and find its intercept OE on a 
line drawn perpendicular to the radius vector OA. By 
carrying out this construction for a number of points and 
marking off values of OE along OA (as at OF, Fig. 26) in 



POSITION, VELOCITY, AND ACCELERATION. 57 

each case, we find points on a new curve, QF^F^, which 
is in fact a polar velocity -time diagram. 

The reader will now be in a position to see that by repeat- 
ing the same construction with the new velocity curve a 
further diagram is obtained, that of acceleration; since 
acceleration has exactly the same relation to velocity that 
velocity has to displacement. In this case, of course, the 
quantit}^ represented by a line drawn in the same way as OE 
above is the rate of change of velocity with regard to time. 

We may next study some examples of such diagrams. 

Fig. 27 represents a polar diagram or curve of dis- 
placement, in which distance increases uniformly with 
time, as shown by the fact that QO-PO=PO-NO, if the 
angle QOP = angle PON, and so on. The curve ONPQ is 
of course an Archimedean spiral. It may be proved that 
the length of the intercept OE by the normal DE on a line 
OE through perpendicular to the radius vector is con- 
stant whatever the position of I^ * on the curve. This 
length OE has been shown to represent the velocity, and 
accordingly in this case the velocity diagram will be a circle 
with radius OE, and the acceleration will be zero, since the 
velocity is uniform. In the drawing, if the displacement 
scale is as shown, and the time scale is 90° = 10 seconds, the 
velocity of the body is i foot per second. To determine the 
velocity scale of such a diagram we need only reflect that 

* The polar equation to the spiral of Archimedes is r = aB ; 

hence 6 = - and — ■ = —. 
a dr a 

If (p be the angle which the tangent makes with the radius vector, 



tan d) = r-—. 
ffy 



r 
Thus tan = — , 
a 



Now the angle between the normal DE and a line OE perpendicular to the, 

y a 

radius vector is 0. Hence OE = :; = r.— = a = constant. 

^ tan (p r 



58 KINEMATICS OF MACHINES. 

ds 
the velocity is numerically -r- . If the displacement scale 

were i inch = 3 feet, a velocity of i foot per second would 
be represented by a length of one third of an inch 
divided by the angle representing i second. In this casfe 




10 FEET 



TIME ON.E SECOND = 0.157 RADIAN 

VELOCITY ^^ \ i^'>8E0. 

Fig. 27. 
90° represents lo seconds of time, so that i second of time 

would be denoted by — or 0.157 radian. Hence unit veloc- 

20 

ity would be represented by =2.12 inches, and the 

.3X0.157 

velocity scale would be i inch = 0.471 foot per second. 

23. Diagrams for Simple Harmonic Motion. — In the 

following chapter we shall frequently meet with cases in 
which 'bodies have periodic motion in a straight line, either 
exactly or approximately harmonic in character. We 
define simple harmonic motion as the motion of a point which 
is the orthogonal projection on a straight line of another 
point moving uniformly in a circle, termed the auxiliary 
circle. The radius of the auxiliary circle is called the ampli- 
tude of the motion. Such motion can be conveniently 
studied by means of polar diagrams; the engineer, for 
example, often employs such diagrams to elucidate the 
action of the slide-valve of a steam-engine, for the valve 



POSITION, yELOClTY, AND ACCELERATION. 



59 



has very nearly such a motion as has been defined above. 
In Fig. 28 let AOB represent the path of a point having 




Fig. 28. 
simple harmonic motion; we wish to draw diagrams of 
displacement, velocity, and acceleration for this point. AO 
is the amplitude of the motion, AmBq is the auxiliary circle, 
and, according to our definition, LMNP, etc., will be posi- 
tions of the point after the lapse of times proportional to 
the angles AOl, AOm, AOn, AOp, and so on. It is plain, 
then, that a polar time-displacement diagram can be readily 
drawn by marking off along each radius a distance equal to 
the corresponding displacement of the vibrating point from 
its mid-position. Thus OU has been made equal to OL. 
It should be noticed that during each revolution of the 
rotating point round the auxiliary circle, its projection 
travels from A to B and back again, so that twice during 
each period the displacement of the vibrating point will be 
zero, while at the same time the velocity will have its 
greatest value. 

The period is of course the time of a complete revolution of 
the point round the auxiliary circle. During one half of this 
time the vibrating point has a positive velocity, i.e., is mov- 
ing in one sense, say from right to left, while during the 
remainder of the period the velocity will have a negative 
value. 



6o KINEMATICS OF MACHINES. 

The displacement diagram is drawn by obtaining a series 
of such points as U , where OU = OL ; the locus of such points 
is easily shown to be a pair of circles touching the auxiliary 
circle and each other. Taking the point L', for example, 
join UA . Then in the triangles OIL, OAL' we have OL' = OL 
and 01 = OA , while the angle 10 A is common to both. Hence 
the triangles are equal in all respects, and the angle OL'A is 
a right angle. Therefore the point U lies on a semicircle 
drawn on OA as a diameter, and the complete locus of L' is. 
a pair of circles, as shown. The radius vector OL' repre- 
sents the displacement of the vibrating point at a time 
represented by the angle between the radius vector and the 
initial line OA . 

Having given such a diagram of displacement (Fig. 29)^ 
let us apply to it our construction for determining velocity. 
At any point D the line DE is drawn normal to the curve of 
displacement, and is cut by OE where the angle EOD is a. 




Fig. 29. 

right angle. Accordingly, when the length OE is marked off 
at E' along OD we get one point on the velocity diagram. 
Plainly, in this particular case, any radius vector of the ve- 
locity diagram, such as OE' , is equal in length to the radius, 
vector of the displacement diagram which makes an angle 
of 90° with it, and it follows that the velocity diagram will 



POSITION, VELOCITY, AND ACCELERATION. 6i 

also be a pair of circles, as shown in the figure. Their axis is, 
however, at right angles to the axis of the displacement 
diagram, the maximum velocity being reached when the 
body is at the middle point of its path. 

On constructing the diagram of acceleration, we find 
that it also takes the form of a pair of circles and coincides 
with that of displacement. The scales of the two diagrams 
are of course not the same, but it follows that in simple 
harmonic motion the acceleration at any instant is propor- 
tional to the distance of the vibrating point from its mid-posi- 
tion, a fact which can also be readily proved analytically.* 

Fig. 30 gives the linear and polar diagrams of displace- 
ment, velocity, and acceleration for a simple harmonic 
motion of which the amplitude is 0.75 foot, while the period 
is ^ second. This corresponds approximately, but by no 
means exactly, to the motion of the piston of a steam-engine, 
18 inches stroke, and making 300 revolutions per minute. 

The linear diagrams have been plotted on a straight line 
base from the radial diagrams ; they, might have been drawn 
by the methods of §§ 18 and 19, in which case their 
scales would have been different from those shown. In 
general, polar diagrams are easier to draw than linear dia- 
grams, and are more useful for cases of periodic motion 
such as this ; the linear diagrams are added for the sake 
of comparison. 

With regard to the scales of these figures, the scale of 
time as originally drawn was 180° =0.1 second (i.e., i second = 

— = 31.416 radians), while that of distance was i foot = i 

inch. Accordingly unit velocity was represented by a length 

of — — 2 inch. The velocity scale was thus i inch = 31.416 
31.416 

feet per second. Unit acceleration was represented by 



* See Perry, Applied Mechanics, p. 549. 



62 



KINEMATICS OF MACHINES. 



I foot per second .0318 • . . ^ 

-. , or ^ = .001011 men, hence the accel- 

I second 31.416 

eration scale was i inch = 988 feet per second per second. 



SCALE OF time: 1 RADIAN = .038 SECOND 




0.05 6EC. 



0.05 SEC. 




0.05 SEC 




DISPLACEMENT 

DISTANCE 
VELOCITY 
\CCELERATION 



ACCELERATION 



■ ^0 . ^0 ' 6.Q 



80 



100 FT. PER SEC. 



1000 



2000 



3000 FT. PER SEC. 
-J- PER SEC. 




DISPLACEMENT AND ACCELERATION 



0.05 



0.1 



0.15 



0.2 



0.25 SECOND 



SCALE OF TIME 

Fig. 30. 



Note that the velocity of the rotating point round the 

I Ktz 
auxiliary circle is ~-^ =23.6 feet per second. The max- 
imum velocity of the vibrating point as measured from the 
diagram is 0.75X31.416 = 23.6 feet per second, thus agree- 
ing with our definition of simple harmonic motion. The 
maximum acceleration of the vibrating point will be found 
to be the same as the radial acceleration of the point 



POSITION, VELOCITY, AND ACCELERATION. 



^Z 



V 



travelling round the auxiliary circle, namely — , or 



= 741 feet per second per second. 



23.6X23.6 

0.75 

If an acceleration diagram on a displacement base is re- 
quired, this can easily be drawn from the acceleration- 
time and displacement-time diagrams, and it will be found 
to take the form of a straight line, since, as has previously 
been remarked, acceleration in simple harmonic motion is 
proportional to displacement. 

24. Relative Motion of Two Bodies each having S. H. M. — 
Cases arise in which it is necessary to find the relative 




Fig. 31. 

motion of two or more bodies having simple harmonic 
motion, or we may wish to determine the motion resulting 
from the combination of two or more simple harmonic 
motions. We proceed to show how this is done. 

It has been seen that every simple harmonic motion is 
capable of being referred to a corresponding uniform motion 
round a circle called the auxiliary circle. At any instant, for 
example, the simple harmonic motion of the point L (Fig. 
31) corresponds to the uniform motion round the circle 
of the point /. 

Let there be two points, L and M, having simple har- 



64 



KINEMATICS OF MACHINES. 



monic motion (of the same or different amplitude), and 
let / and m be their reference points. If AOB is the line 
on which the motion of I and m is projected, we define the 
constant difference between the angles COl and COm as 
the difference of phase in the two simple harmonic motions. 
In Fig. 31 the point M has a motion differing in phase by 
— 90° from that of the point L ; in other words, Om con- 
tinually lags 90° behind 0/, and of course M lags behind L 
to a corresponding extent. 

In Fig. 32 let there be two points, L and M, having sim- 
ple harmonic motions along AOB, whose amplitudes are 




Fig. 32. 



01 and Om, and whose difference of phase is the angle lOm, 
the period being the same for both motions. We wish to 
find the relative motion of L and M, compounded of their 
two simple harmonic motions. 

Join Im and draw On equal and parallel to ml. Draw. 
nN perpendicular to AB. 

Then the relative displacement of L and M is OM — OL, 



POSITION, VELOCITY, AND ACCELERATION, 



65 



where OM or OL is reckoned negative when measured to 
the left of 0. Now OM -0L= ML is the projection on AB 
of the line ml. Also, ON is the projection on AB of the line 
On, equal and parallel to ml. Thus ON =ML, since each 
is equal to On cos NOn. 

Hence in any position the relative displacement of M 
and L is equal to the distance of the point A^ from the cen- 
tre 0. But N has a simple harmonic motion of the same 
period as that of M and L, of amplitude On =ml, and differ- 
ing in phase from that of L by the angle lOn. Thus if two 
points have simple harmonic motions of the same periods 
and along the same straight line, their relative motion is 
also a simple harmonic motion, in general differing in ampli- 
tude from either of the components, and also differing in 
phase. 

In Fig. 33 diagrams of displacement are drawn for two 
simple harmonic motions of 2 seconds period, the amplitudes 
being J inch and i inch, and the phase difference 60°. 



/2 SEC. 




3 INCHES 



The curves U and M' have ordinates proportional to 
the displacements of the corresponding points along AB, as 
represented by the motion of the points PQ round their 
respective auxiliary circles. The resultant displacement 
of L relatively to M is shown by the distance M'U =0'N\ 
N' being above or below the line CD (along which time is 
measured) according as L is above or below M along the 
lineA^. 



66 



KINEMATICS OF MACHINES. 



Thus the curve N' is drawn through points whose heights 
above CD are equal to the heights of L above M at different 
times. It will be found that these displacements are the 
same as those of a point N moving along AB with a simple 
harmonic motion of period 2 seconds, amplitude 0.9 inch^ 
whose phase is about 45° behind that of L and 105° behind 
that of M. 

The curves drawn in Fig. ;i,T^ are in fact sine curves^ 
since their ordinates are proportional to the sines of angles 
which are proportional to the abscissas. In general, on 
compounding simple harmonic motions of different periods 
we do not obtain a simple harmonic motion as a result, but 
a more complex movement which is still, however, periodic- 
A case of this is shown in Fig. 34, where two simple har- 




FiG. 34. 



monic curves, A and B (shown by dotted lines), differ- 
ing in phase, amplitude, and period, are compounded, the 
resultant curve C being shown by a full line. It is possible 
to resolve the curve representing any periodic function into 
a number of component sine curves of different phase, ampli- 
tude, and period. In this way, for example, the complex 
curve drawn by a tide-gauge is analyzed, and the periods, 
amplitudes, and phases of its component tides are deter- 
mined. 

In the figure, for instance, the curve CC might represent 



POSITION, VELOCITY, AND ACCELERATION, 



67 



the actual rise and fall of the water-level at a certain place, 
due to the simultaneous effects of two tides, which, acting 
alone, would respectively produce the fluctuations shown 
by A A and BB. Distances measured horizontally repre- 
sent time as before. 

Such diagrams as those given above enable us to study 
any periodic function, and they find many important ap- 
plications in scientific work. 

25. Composition of S. H. M. not along Same Line. — Imag- 
ine that a point, having simple harmonic motion in the 
direction of a given line AB, has impressed upon it an- 
other simple harmonic motion in the direction of a line CD 
at right angles to AB. The motion of the bob of a simple 
pendulum is very approximately simple harmonic. An 
arrangement like that shown in Fig. 35 may be devised, 
which will give to a pencil, P, a motion compounded of 
those of two pendulums, Q and i?, swinging in planes at 
right angles to one another. 




Fig. 35. 



It is easy to see what the path of such a pencil will be. 
In Fig. 36 let 0123456 represent successive positions of the 



68 



KINEMATICS OF MACHINES. 




'---.5 



\4 



>2 



12 3 4 5 6 




PERIODS EQUAL, AMPLITUDES EQUAL 




a, SAME PHASE 




5. PHASE DIFFERENCE aO"* 




C, " " 9.0» 




Fig. 36. 




3 


^-^^ 



1/ 



X5 



/< 


ri 


'"^vl 1 ^ >.^1 1 """^vl \ 


/" 






/8 

1 






h6 




__| _! M<-^ \i-"' 1 


1 




i ' -^"^ ' '^<rci"\.'/ 




._( -_^ .^-_-,^ ^ -^ yf-c 


\ 




1 ,^ 1 ^^'' 1 ' >r 1 \ 


V 

\^ 2 . 






"<^ 


1 

-4. 






PERIODS 2:1, AMPLITUDES. EQUAL 






a, INITIAL PHASE DIFFERENCE 0° 






6, " " " 60° 






C, " " " 90° 



Fig. 37. 
vibrating point measured along one line, and o i'2'3'4'5'6' 
those measured along the other, equal intervals of time 
being taken, and the periods of the two motions being equal. 



POSITION, VELOCITY, AND ACCELERATION. 



69 



In the figure the movement from o to i or i to 2 is executed 
in yV of a complete period. Now if both harmonic motions 
have the same phase, points of intersection of the Hnes 
drawn through i and i', 2 and 2', 3 and 3', etc., parallel to 
the axes OA, OC, will give successive positions of the trac- 
ing-point.^ If there is a phase difference of 60°, the tracing- 
point will have moved as far as 2' along one line, while it 




3 6 9 12 15 18 

PERIODS 2:3, AMPLITUDES EQUAL 
a, INITIAL PHASE DIFFERENCE 0° 
h, " " " 30° 

Fig. 38. 

is still at o on the other ; hence the imes drawn through i 
and 3', 2 and 4', and so on, now give the path, which is 
seen to be an ellipse. Again, with a phase difference of 
90° the path becomes circular. Figs. 37 and ^% show the 
curves resulting from the compounding of two simple 
harmonic motions of equal amplitudes, having periods in 
the proportion of 2 : i and 2 : 3 respectively, and various 
initial phase differences The combination of motions 
which have periods in any other ratio can readily be illus- 
trated by the same method, and the reader will find it in- 
structive to plot for himself some of the resulting curves. 



CHAPTER III. 

PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 

26. Quadric Crank - chains. — If we endeavor to make a 
plane mechanism out of links containing only turning pairs, 
we find that the least number of links with which this can 
be done is four. A chain of three links so connected forms 




Fig. 39. 

an arrangement which is of value as a structure (a simple 
triangular roof - truss) , but is of no service as a mechanism, 
since its parts can have no relative motion. 

On the other hand, a simple chain of five or any greater 
number of links connected by turning pairs is equally useless 
as a mechanism, since the relative motion of at least two 
of its links is not constrained, as has been shown in § 3. 

Let us consider, therefore, a chain of four links connected 
by turning pairs whose axes are parallel. When the links 
of this chain are of unequal lengths the smallest is called the 
crank, and since the four links form a quadrilateral, the 
chain has been called by Reuleaux * the quadric {cylindric) 
crank-chain. The term ' cylindric ' distinguishes this chain 

* Kinematics of Mach., §§ 62-65. 

70 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 1i 

from the corresponding spheric chain, in which the axes are 
not parallel. 

In quadric crank-chains it will be convenient to dis- 
tinguish between links having a swinging or partial turning 
movement and those which can execute complete rota- 
tions relatively to the fixed link in the chain. 

The former links will be called levers, the latter cranks. 
It is obvious that by altering the relative lengths of the 
links we can obtain different relative motions, and hence 
different mechanisms. From these, again, other different 
mechanisms are produced by inversion of the chain. 

27. Virtual Centres and Centre des. — Let ahcd, Fig. 40, 
represent the four links of a quadric crank-chain. Each 
of these links will have motion relatively to every other, and 
hence we shall have six virtual centres. Four of these cen- 
tres are readily identified as the axes of the turning pairs; 
for instance, the virtual centre of c with regard to <i, or of (i 
with regard to c, is obviously the point 3, and may be 



Oac ^--• 




indicated as 0^^ or 0^^. In the same way we have 0^^, j, 
and Oq^; all these points are in fact permanent centres as 
regards their own pair of links. Remembering that for any 
three bodies having plane motion the three virtual centres 
lie in one straight line, it is easy to see that 0^^ must lie at 



72 KINEMATICS OF MACHINES. 

the join of the straight lines drawn through 0^^ and 0^^^ 
and through 0^^ and 0^^. In the same way 0^^ is at the 
intersection of the lines O^fi^^ and ^fi ^^. 

Supposing h to form the frame or fixed link, it is seen 
that since 0^^ and 0^^ are permanent centres, the centrodes. 
of a and c with regard to b are points, namely 0^^ and 0^^. 




Fig. 41. 

The centrode of d with regard to b is the locus of O^f^ and 
takes the form of a curve having four infinitely distant 
points; portions of it are readily drawn by finding a 
series of positions of O^j, corresponding to successive posi- 
tions taken up by the three links a, c, and d. In a similar 
way may be obtained the centrode of b with regard 
to d (supposing d to be the fixed link). The curve PQRS 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 73 

in Fig. 41 represents the centrode of d with regard to h ; the 
construction for two points on the curve is shown. 

28. Angular Velocities. — It is frequently of importance, 
having given the angular velocity, say, of the link a, to find 



02 4 G 8 10 12 FEET 

1 ! 1 1 I I I I I I I I I 

P 2 4 6 8 10 RADIANS PER 8Ea 

l \ I I I I 1 ■ I ' 1^ 




Fig. 42. 

that of any other link, say c, or, in other words, to determine 
the angular velocity ratio of the chain. This can be very 
simply done by construction. 

In Fig. 42 let A BCD represent the mechanism, AD being 



74 



KINEMATICS OF MACHINES. 



the fixed link, and the uniform angular velocity of CD being 
known. It is required to determine the angular velocity 
oi AB for any position of the mechanism. 

Draw DE parallel to AB, and cutting BC, or BC pro- 
duced, in E. With centre D and radius DE mark off DF 



G(=0d5; 




Fig. 43. 

along DC, Then DF represents the angular velocity of c 
on the same scale as that on which AB represents the 
angular velocity of a, and if a series of points such as F 
be obtained, the curve FFJ^fi . . . drawn through them 
will form a polar diagram of angular velocities for c and a. 

To prove this construction, let w^^, (o^^ be the angular 
velocities of c and a respectively with regard to h. In Fig. 
43 find 0^^, the intersection oi AB and DC at (J, and draw 
DE parallel to AB, meeting BC in E. 

Since the link d is turning for the instant about G, we 
must have 

linear velocity oiB _ GB 
linear velocity of C GC 



Now 



^.6 = 



linear velocity of B 



AB 



and 



ab 



linear velocity of C 
CD 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 75 

(o . linear velocity of 5 CD 
(o^f^ linear velocity of C A -B 

CD.GB 
~AB.GC' 

But by construction the triangle BGC is similar to the 
triangle EDC ; hence 

GB^ED 
GC~DC 

oj, CD. ED ED 

Therefore _ ^ = = 

inereiore AB.DC AB 

ao 

Thus ii AB represents the angular velocity of a with 
regard to b, ED represents on the same scale that of c with 
regard to b. 

Fig. 42 gives such a velocity diagram, drawn to scale, for 
the beaiTL of a beam-engine when the crank rotates uniformly. 
For comparison the circle of radius DH = AB has been drawn, 
so that for any radius DGH the intercept DG represents 
w^f^, just as DH represents w^^. The polar curve of velocity 
is shown by a dotted line. 

The distances taken are : 

AB = Sieet=DH; 
BC = 20 ieet; 
CD = 4 feet ; 
DA =21.5 feet. 

When the crank is in the position DH the angular 
velocity ratio is 

DG 3-5 



DH 8 



0.438, 



or at that particular instant the beam is swinging with 0.438 
the angular velocity of the crank. If the crank rotates 



76 KINEMATICS OF MACHINES. 

uniformly at 60 revolutions per minute or 6.28 radians per 
second, in the position AB the beam is moving with an 
angular velocity of 6.28X0.438 = 2.75 radians per second. 

From the curve of angular velocity thus obtained we 
might draw the curve of angular acceleration by the con- 
struction described in § 22. Notice that the construc- 
tion just described can still be applied in positions of 
the mechanism where 0^^ is inaccessible, i.e., when AB and 
CD are nearly parallel, and when the relative angular 
velocities, therefore, could not be found from the position 
of the virtual centres. 

29. Inversions of the Quadric Crank-chain. — In the par- 
ticular example of the quadric crank-chain just examined, 
the lengths of the links are such that while the link a exe- 
cutes complete rotations with reference to b or d, c only 
swings, a is then a crank, c sl lever, and if the link b is the 
fixed one, the resulting mechanism is called the lever crank- 
chain. 

In order that a may execute complete rotations with 
regard to b it is necessary that a + b ^c-\-d^ while also 

a + d^c + b, a being the smallest of the links. 

With these proportions let us see the result of inversion 
of the chain. On considering the relative motions of the 
links we find that the motion of a relatively to 6 or J is that 
of complete rotation, while with regard to either of the 




Fig. 44. 
same links c only swings or performs partial revolutions. 
As has been already pointed out, inversion can make no 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 77 

change in the relative motions of the Hnks, and hence the 
mechanism will remain a lever crank-chain whether h ov d 
be the frame (or fixed link) . 




Fig. 45. 



On fixing a, however, as in Fig. 45, a new mechanism 
is obtained which may be called the double crank, inasmuch 
as both h and d can now execute complete rotations about 
the axes of the pairs ha and da. 

This mechanism is used in practice as a drag-link coup- 
ling, h and d being represented by the discs keyed on to 
the two shafts, a by the frame containing the bearings, 
and c by the drag-link connecting the pin on d with that 
on h. The mechanism is also employed in the construction 
of feathering paddle-wheels. 

When used for this purpose the object is to cause the 
floats to enter and leave the water edgewise, (so as to avoid 
splashing,) while remaining vertical at the bottom of their 
travel. 

Thus suppose A,B,C, Fig. 46, to be points on the path 
of the outer edge of a float, AC being the water-line. The 
steamer has a certain speed relatively to the water, so that 
a point A on the wheel is moving horizontally with a velocity 



78 



KINEMATICS OF MACHINES. 



AD in common with every other point on the vessel, the 
length oi AD thus representing the speed of the ship to any 
convenient scale. In virtue of the rotation of the wheel, A has 
also a linear velocity (represented hy AE to the same scale), 
relatively to the ship ; therefore the real direction in which 




10 FEET 



1000 



2000 



3000 



4000 FT. PER MIN. 



Fig. 46. 



A moves relatively to the water is AF, the diagonal of the 
parallelogram AEFD. The floats then at entrance and 
exit should lie in the positions AF and CH, while the float 
in its lowest position should evidently be vertical, as RtBG. 
The floats are pivoted at their centres to the framework of 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 79 

the wheel and have attached float-levers KL, MN, OP. 
The ends L, N, and P are all connected by radius-rods to an 
eccentric -pin, generally fixed on the sponson-beam of the 
paddle-box. The centre of this pin is of course at Q, the 
centre of the circle passing through A^, P, and L, while i? 
is the centre of the wheel itself. It will be seen that the 
paddle-wheel arm, the float-lever, the radius-rod, and part 
of the ship's structure form a double-crank mechanism, 
thus giving the floats the desired movement. Fig. 46 is 
drawn to scale from the following data : 

Speed of ship 21 knots = 2026 feet per minute. 

Diameter of wheel (ext.) 18.5 feet. 

Revolutions per minute 48. 

Breadth of floats 3' o''. 

Immersion of lower edge 3' 6". 

Length of float-levers 3' o''. 

Speed of outer edge of float 2 790 feet per minute. 




Fig. 46^?. 

Fig. 46a is a drawing of the arrangement of an 
actual feathering paddle-wheel. The reader will have 



So 



KINEMATICS OF MACHINES. 



no difficulty in recognizing various links in the double-crank 
mechanism. 

Next suppose c (Fig. 44) to be the fixed link. Remem- 
bering that the relative motions of h and c and d and c are 
partial and not complete rotations, we see that a third 
mechanism, the double lever, is the result of this inversion. 




Fig. 47. 



The double -lever mechanism is shown in Fig. 47, and 
such an arrangement finds an application in certain approx- 
imate straight-line motions (compare Fig. 566). 

The result of the various inversions of the quadric crank- 
chain may be summarized as follows : 

Fixed Link. Mechanism. 

a . .Double crank. 

h .Lever-crank. 

c Double lever. 

d. : Lever-crank. 

The four inversions have thus given us three different 
mechanisms. 

Certain special cases of the quadric crank-chain have 
peculiarities which are of interest. Suppose that the 
lengths of the links are such that either a-\-h=c-\-d or 
a-{-d = c-\-h, h being the fixed link. This condition is shown 
in Fig. 48. 

As we have seen, it is still possible for a to execute 
complete rotations, but it will now be found that c can swing 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 8i 

on either side of the fixed link h. We obtain one position of 
the chain in which all the links are in a line, and the angle 
through which c can swing is then doubled. This condition is 
expressed by saying that the mechanism passes through 





---:^-' 



Fig. 48. 



a change -point when all the links are m line, and for any 
given position of the link a the mechanism can assume 
either the position shown by the full lines or that shown 
by the dotted lines in the figure. 

30. Change-points and Dead-points. — A change-point 
may be defined as a position of a mechanism in which such 
a want of constrainment occurs that it is possible for the 
arrangement to transform itself into another mechanism, or, 
in some cases, into a pair of elements. 

A very familiar instance of such a change-point occurs 
in the quadric crank-chain when of the form known as 
parallel cranks; that is, when a and c are equal in length, 
and considerably shorter than h and d, which are also equal 
This is of course a particular case of the condition illus- 
trated in Fig. 48, and is shown in Fig. 49. 



82 KINEMATICS OF MACHINES. 

At the instant when all the links are in a line it becomes 
possible for the mechanism ahcd either to take up such a 
form as ahc^d' or to continue its motion in its original form. 
The mechanism in question is of common occurrence in 
locomotive engines having coupled driving-wheels, and the 
necessary constraint at the change-points is provided by 
duplicating the chain, namely, by arranging another pair 
of cranks and a coupling-rod on the other side of the engine, 
so placed that the change-points of the two chains do not 
occur at the same time. Other methods of obtaining a 
similar object will be found discussed in a later chapter. 




Fig. 49. 

By the term dead-point in a mechanism is meant a 
position of the various links such that one of them directly 
opposes itself to the action of the forces tending to pro- 
duce motion. The term was first applied by Watt to those 
positions of the crank and connecting-rod in a steam-engine 
in which the axes of three turning pairs lie in one plane, 
so that a force applied to the piston is not able to cause any 
torque on the shaft. It is plain that, in the absence of some 
means of overcoming this difficulty, the further motion of 
the chain becomes impossible, and the chain may be re- 
garded as incomplete. The occurrence of dead-points must 
not be confused with that of change-points, although they 
may, and often do, occur together. 

It is important to note an essential difference between 
dead-points and change-points. The occurrence of a dead- 
point depends on the particular link to which the driving 
force is applied, and on the manner of its application. For 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 83 

instance, in the lever-crank mechanism of Fig. 42, if the 
crank be turned by the application of a continuous torque 
to its shaft, no dead-point exists. In the very same 
mechanism, however, if the driving effort be applied to 
the lever c (as in a beam-engine), dead-points occur twice 
in each revolution of the crank. 

A change-point, on the other hand, is caused by the con- 
figuration of the chain itself, and is' present whichever link 
is fixed, so long as the chain is the same. 

31. Special Forms of Quadric Crank-chain. — Suppose 
in the quadric crank-chain that we make a = 6 and c = d\ we 
then obtain a chain called by Professor Sylvester the kite, 
from its form. When one of the links is fixed, and the 
motion examined, it will be found that there are two change- 



FiG. 50. 

points. Thus, in Fig. 50, imagine that a rotates in the direc- 
tion of the arrow, h being fixed. When the joint 4 coincides 
with 2 the chain becomes, for the instant, a turning pair, 
having its centre at 2, c and d rotating together. If a con- 
tinues its motion for another complete rotation, another 
change-point occurs, the chain having passed through the 



84 KINEMATICS OF MACHINES, 

positions i, 2,6, 7; i, 2, 8, 9; i, 2, 10, 11; and so on, until 
the position i, 2, 12 is reached. If proper restraint is 
appHed at the change-points, so as to prevent c and d rotat- 
ing together, we see that for one complete rotation of c, a 
will have made two revolutions. 

The kite finds its principal application in certain 
straight-line motions,* some of which will be presently 
discussed. 



L_ 


in 








n\ 


/ 


»A 










^ 


/ 


1 ^o>~-^^ 




























/ 


'^x-O^ 




ip 


A' 












/ 


'^^"- 


J 


F 








' 






/ 


\ 1 




-I::! 


t-~. 



c 



B' 



///////////////// 




Fig. 51 

A quadric crank-chain in which opposite links are of 
equal lengths is employed (with the addition of a fifth fixed 
link) for copying purposes iinder the name of a pantagraph . 

Let OABC, in Fig. 51, represent such a chain attached 
to a point on a fixed link at 0, and let E and D be any two 
points fixed on AB and CB, so that OED is originally a 
straight line. 

* Kempe, How to Draw a Straight Line. Macmillan, Nature Series, 1877 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 85 

The figure OABC is a parallelogram whatever the posi- 
tion of the mechanism; hence the angles OAE, EBD are 
always equal. The lengths of the links are invariable, 

hence — p. = -5-^ fo^ any position, so that the triangles OAE, 
AE BE 

DBE are always similar, the angle OE A = the angle BED, 

and OED remains a straight line whatever the position of 

the mechanism, as at OA'B'C\ 

If we suppose a pencil to be attached to E, any figure 

it describes ' will therefore be the polar projection of the 

figure described by D; the two curves being similar and 

similarly placed with regard to the pole 0. The ratio of 

OE 
reduction is of course -^Yn' ^^^ fixed point need not be 

at the join of the two links,, but may be anywhere on any 
link, so long slsO, E, and D are taken in one straight line. 

If in the ordinary parallelogram we replace the bars AB, 
BC by wider pieces of material, and then choose on these 
pieces points P and Q such that the triangles PA 5, BCQ are 
similar, we obtain a mechanism called by its inventor, 
Professor Sylvester, the skew pantagraph (Fig. 52). 



,R jaP 



mmm 

Fig. 52. 

Join OP, OB, OQ. Then 

PA_BC . P^_OA 

AB'CQ OC~CQ" 



86 



KINEMATICS OF MACHINES. 



But the angles PAO, QCO are seen to be equal, there- 
fore the triangles PAO, OCQ are similar and 
OP OA AP ^ ^ 

Further, the angle POQ^AOC-AOP-QOC 

==AOC-{AOP-\-APO). 
Produce OA to R ; then 

angle POQ =^RAB- RAP =^PAB= constant. 
From the facts that OP and OQ are in constant ratio 
and include a constant angle, it is evident that the paths of 
P and Q are similar but of different sizes, and one is turned 
through the angle POQ with regard to the other. 




Fig. 53. 
The parallelogram linkage is also applied in other ways. 
Fig. 53 represents the Roberval Balance in which a fifth 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 87 

link is added and the parallelogram is thus doubled. Dur- 
ing the movement of the mechanism the distance moved 
by a point on c is equal to that moved in the opposite sense 
by any point on a. Accordingly if used as a balance it 
does not matter where the weights are placed on the scales, 
and the latter remain always horizontal. 

32. Straight-line Motions. — Under the name straight- 
line motions may be considered mechanisms whose charac- 
teristic feature is that one or more points in them travel 
exactly or approximately in straight lines, without being 
directly guided. Those contrivances which contain turn- 
ing pairs only will be discussed here ; nearly all of them are 
formed from the quadric crank-chain. 

In the earlier days of mechanical engineering plane sur- 
faces were not so readily produced as at the present time, 
and straight-line motions, or parallel motions as they were 
called, played an important part as substitutes for straight- 
line guides. The linkage employed by Watt is shown in its 
simplest form in Fig. 54, and is seen to be a quadric crank- 
chain so proportioned that one point on the shortest link 




>Oac 



describes a path of which certain parts are approximately 
straight, the whole path being a kind of lemniscoid, or figure 
of eight. Such a mechanism is now used as a guide for the 
motion of the pencil of a Richards steam-engine indicator. 

The mechanism consists of four links, the fixed link 
c being usually the longest, while h and d are generally of 
equal length. 



88 KINEMATICS OF MACHINES. 

This condition is not, however, necessary. Suppose 
the proportions to be those shown in the figure. When the 
mechanism is in such a position that h and d are parallel, 
as shown by dotted lines in the figure, the virtual centre of 
a with regard to c is at an infinite distance away, in a 
direction parallel to h and d. At that instant every point 
on a is moving in a direction perpendicular to its virtual 
radius and therefore at right angles to AD and CB. We 
wish to find a point on a which will describe a straight line 
(approximately) during small displacements of the mechan- 
ism from its dotted position. Such a straight line must 
be at right angles to the parallel position of h and d, as we 
have just seen. 

Imagine that the mechanism is moved into such a posi- 
tion as that shown by the full lines AD'C'B, h describing 
a small angle Ad, while d describes a small angle A (p. The 
position of the virtual centre 0^^ being found at 0, it is evi- 
dent that the point required on CD' must be such that its 
virtual radius is parallel to AD or CB, for if not, it would be 
describing a line not perpendicular to those lines, and there- 
fore not in a straight line with its original direction of motion. 
If, then, we draw a line OE parallel to AD and cutting CD' 
mE, E \s the point required. Draw C'F, D'G, perpendicu- 
lar to OE. For an indefinitely small displacement of the 
mechanism the link a may be considered as remaining sen- 
sibly parallel to itself, so that DD' = CC and 

AD _A(p^ 
~BC~~Jd'' 

also for such a displacement is at so great a distance that 
we may write OG = OF without sensible error ; hence for 
very small displacements 

Ad D'G~WE' 



Finally, o^-n/zr' 



AD _CE 
~^C~WE' 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 89 

SO that E must divide CD' in this proportion if its path is 
to be a straight hne for small displacements. For larger 
displacements the path departs considerably from such a 
line. 

It is better to proportion the lengths of the links so that 
DC is vertical when AD and BC are parallel ; in that case 
the path of E will be approximately straight over a larger 
range of movement. 

The links h and d may be placed both on the same side 
of a, as in Fig. 55. In this case E will be found to lie out- 




FiG. 55. 



DE TiC 
side CD, but as before — — ^ = -— -. The figure shows one 

CE AD 

half of the complete path of the point E, as well as two 

positions of the mechanism. 



90 KINEMATICS OF MACHINES. 

This arrangement was also used by Watt. 

It may be shown that the part of the path of E used for 
a straight line is in reality wavy. Rules for designing Watt 
straight-line motions, as well as other forms, are given in 
Rankine's " Machinery and Millwork," § 253 ^^ seq. 

A number of other approximate straight-line motions * 
are derived from the quadric crank-chain. Among these 
the most interesting are those of Roberts and of Tchebi- 
cheff ,t shown in Figs. 56a and 566. 





ROBERTS ' TCHEBlCHEFF 

Fig. 56a?. Fig. 56<5. 

33. Accurate Straight-line Motions. — The first geomet- 
rically correct straight- line motion was devised in 1864 by 
M. Peaucellier. It is shown in Fig. 57 and is a compotind 
chain of eight links of which 

a = b=c = d; 

g-h. 

The links a, 6, c, d, e, f form a kite-shaped figure known as 
the Peaucellier cell. It has the following properties : 

(i) Ay D, and C must always be in one straight line. 

(2) AB'-BC'=AG'-GC' (since AC and BE are at 
right angles). 

Hence AB'-BC' = (AG + GC)(AG-GC)= AC. AD. 

Therefore ACAD is constant. 

* Burmester's Lehrbuch der Kinematik, § 255 et seq. 
f See Engineerings Vol. XVI, p. 284. 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 91 

Suppose the chain moved until the line AC coincides 
with AF. Let / be the position of C, while H is the posi- 
tion of D. We have 

HA.JA=DA.CA, 
HA CA 
DA-JA' 
Also, FH=FD=FA, so that H, D, and A lie on a semi- 
circle. 





Fig. 57. 



92 



KINEMATICS OF MACHINES. 



But in the triangles HAD, CAJ the angle CAJ is com- 
mon, so that the triangles are similar, and the angle CJA is 
equal to HDA, which is the angle in a semicircle. There- 
fore CJA is a right angle. In the same way it may be 
proved that for any other position of C the line drawn from 
C to J is at right angles to A J. 

Hence all possible positions of C lie on a line drawn 
through / perpendicular to A J, and therefore C describes 
a straight line. 

Let us inquire what is the result of varying the lengths 
of the links g and h. It is plain that by making h=o the 
points A and F would coincide and C would describe a 
circle of radius AC. From this it would appear probable 




Fig. 58. 

that by giving h suitable lengths less than that of g, we 
might describe circular arcs of any radius from AC wp to 
infinity. If h be made greater than g, c will still describe 
a circular arc, but it will be convex towards A and the 
radius continually diminishes as g diminishes. 

The locus of C may be proved to be a circular arc as 
follows : 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 93 

In Fig. 58 we have, as before, e'^-a^=AC . AD. On FA 
produced mark off AO=k, where 

._-Ke^-a^) 
and 

Join OC. Draw OK, FL perpendicular to AC. Let the 
a,ng\e CAF = a. Then 



AD = h cos a + Vg^ — h^ sin^ a^ 
from which 



AD' + h'-g' 
^"^"^ 2h.AD ^^) 



Again, 



AC = VOC^ — k^ sin^ o[ — k cos a, 

from which 

OC'-k'-AC , . 

^"^"^ 2k. AC ^'^ 

From (i) and (2), 

AD h'-g' _ OC k AC 

h ^h.AD ~k .AC AC k ' 

h'-g' AC 



But 



Hence 



h.AD k 

AD _ OC'-k^ 
~ir~ k.AC 



and OC» = -fe^-^(ft'-g^)=^-J. 



94 KINEMATICS OF MACHINES, 

But ' ^ ^'-^^ 



Thus 0C=— ^-V^ =^, which is constant. 

/^ -^' 

The locus of C is therefore a circle of radius 

whose centre is at a distance k from A , such that 

h(e'-a') 



k== 



h'-g' 



Note that if h=g, k = oo and R = oo ; the locus of C is 
then a straight line as in the Peaucellier straight-line motion. 
lih<g and ^ > a, i? is positive ; while iihyg,R has a negative 
value, and the circle is convex towards A. The mechanism 
may thus be used conveniently for describing arcs of large 
circles. A graphic method of determining the proper 
lengths of links for this purpose has been devised by Pro- 
fessor Elliott.* 

Another compound chain containing only turning pairs, 
and giving a geometrically correct straight-line motion, 
is that of Bricard.t 

It consists of six links, arranged as in Fig. 59, such that 
the lengths 

AB=^AC = a\ 
FB=GC = b; 
FG = c] 

BD=CE = %-. 
b 



* MacLay, Mechanical Drawing, § XLII. 
f See Comptes Rendus, Vol. 120. 



PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS. 95 

If the point A is to describe the straight hne AK, bisect- 
ing FG at right angles, the length DE must be constant and 
ac 



equal to 



h ' 



This is seen to be the case from the following consider- 
ations: If AF, AG, AD, AE, be joined, it is evident from 




Fig. 59. 

symmetry that the triangles ABF, ACG are equal in all re- 
spects, so that the angles DBA, EGA are equal. The sides 
AB, BD are equal to the sides AG, GE; thus AD=AE. 
Then the triangles ADF, AEG are equal and the angles 
DAF and EAG are equal. Add to each the angle DAG, 
then the angles FAG, DAE are equal. But DA =EA and 
FA =GA, therefore the triangles FAG, DAE are similar and 
DE AD ^^ ^^ AD 

FG^AF^ "^ ^^^^^-ZF 



But in the triangles AFB, 



BAD we have the angle FBA common, while by construe- 
,. FB b AB ,, 
'''''' AB=a==BD'^'^''' 



FA FB . j^j^ ^^ AB ac 

-TTT = -.^ and DE = FG .— — = — . 
AD AB FB b 



96 KINEMATICS OF MACHINES. 

For the required path of A, therefore, DE is constant and 

equal to -X^. 
b 

Further information on the subject of straight-line 

motions will be found in the books and papers to which 

references have been given. 



CHAPTER IV. 
SLIDER-CRANK CHAINS. 

34. Slider-crank Chain. — ^A very important chain is 
obtained from the quadric crank-chain by substituting a 
sliding pair for one of the turning pairs. It is obvious that 
the links will undergo the same relative change of position 
in Fig. 60 (b) as in Fig. 60 (a) , although the lever c has been 
replaced by a block sliding in a circularly curved slot of 
the same radius as the original lever. The chain as thus 
transformed may be called a cylindric slider-crank chain, 
although this name is generally applied to the particular 
case in which 0^^ is at an infinite distance and the block 
slides in a straight slot. It is plain that the mechanism 
of Fig.6o (<:) maybe obtained from that of Fig. 60 (b) by con- 
tinually increasing the radius of the pair cd until it becomes 
infinite. The pair cd may have prismatic surfaces of any 
form so long as the sliding motion is properly constrained; 
thus, for example, c may be a hollow block sliding on a 
prismatic rod J, Fig. 60 (c). The slider-crank chain in its 
cylindric form has of course plane motion, and is of special 
importance, since its different inversions form amongst 
others the mechanisms of various types of reciprocating 
steam-engines. 

The six virtual centres of the slider-crank chain are 
easily found, exactly as in the case of the quadric crank- 
chain, but 0^^ is always inaccessible. Fig. 61 shows the 
centrodes of the links b (representing the connecting-rod 
of a direct-acting engine) and d (representing the frame or 
bedplate). The centrode of b with respect to d (i.e., if d 

97 



98 



KINEMATICS OF MACHINES. 



is considered as the fixed link) is shown by the full line ; the 
dotted curve represents the centrode described by 0^^ if 6 




Ocd 




Fig. 6o. 



is taken as the fixed link. The construction for one point 
is shown in each case. 

AVhen <i is fixed the link c represents the piston, piston- 
rod, and cross-head of the same machine. The link a repre- 
sents the crank, and h the connecting-rod. A point on the 
link h between A and B describes an oval curve with refer- 



SLIDER-CRANK CHAINS. 



99 



ence to d, the shape depending on the position of the point 
selected, and on the ratio of the lengths of crank and con- 





d FIXED 



Fig. 6i. 

necting-rod. This fact is utilized in the design of certain 
valve-gears. 

35. Displacement, Velocity, and Acceleration of Cross-head 
in Direct-acting Engine. (First Inversion of Slider-crank 
Chain.) — One of the most important problems in con- 
nection with the slider-crank chain is the determination of 
the velocity and acceleration of the link c, Fig. 60, suppos- 
ing d to be fixed, and a to rotate with uniform angular veloc- 
ity. This is approximately the case in a direct-acting steam- 
engine, where c would represent the cross-head and h the 
connectin2:-rod. 



3Ctmg- 

L.ofC. 



lOO 



KINEMATICS OF MACHINES. 



It is in general most convenient to deal with these prob- 
lems graphically, but we shall first give an analytical inves- 
tigation. 




Fig. 62. 

In Fig. 62, suppose the line of stroke AO to pass 

through C, the centre of the crank-shaft. Let BC (the 

AB 
throw of the crank) =r, and let -— — =:^, so that the length 

BC 

of connecting-rod =^r = ^5. When the crank makes any 

angle 6 with the centre line AC, let x be the distance of 

the cross-head A from 0, the middle of its stroke. Draw 

BD perpendicular to ^C, and mark off AE = AB. 

the angle of obliquity of the connecting-rod, 

sin 6 

sm (f = 

n 



If 



(f IS 



and 



cos <p = -y/n^ — s>m.^ 6, 



n 



Now 



x=AC-OC=AC-AB 
= CD-VDA-AB 

= r cos 6 + nr cos (p — nr 



= r (cos d — n + \/n^—sm^ 6) (i) 

This gives x in terms of the crank angle d. It is to be 

noticed that when d =— the cross-head is not at the middle 

2 

of its stroke, but at a distance 



XQ = r(Vn^— I — n) 
r 



Vn^—i-\- 



n 



SLIDER-CRANK CHAINS. loi 

the negative sign indicating that A is now to the right of 
0, Fig. 62. 

In the case of a cross-head having simple harmonic mo- 
tion we should have simply 

x = r cos d. 

The term r{\/n^ — sm.'^ d — n) in equation (i) thus gives 
what is called the "error due to obliquity " of the connect- 

ing-rod. Its values for 6 =— are shown below for some usual 

values of n. 

n= 4 5 6 

\/n^ — sm^d — n=—o.i^ — o.ii —0.09 

The error due to obliquity is thus seen to diminish rap- 
idly as n increases.* 

Next, to determine the velocity of the piston at any 
instant we differentiate x with regard to time and obtain 

40 1,0 . , ..-^d 



dx 
dt 



s>md-—A--(n^—sm.'^d) -^ (n"^ — sm.^ 0) \ 
dt 2 ^ dr ^J 



ddr . _ 2 sin <9 COS /9 "I 
dtL 2\/:^'-sin'^J 

This is not very convenient for use in practice, but for 
ordinary values of n we may write without large error n 

instead of \^n^ — sin^ 6. For example, if ^ =-, and sin 6 has 

2 

its greatest value, 

Vn^ — sin^ 19 = 3.87 4.89 5.91 
when n= 4. 5 6 

Further, we may write V^, the linear velocity of the crank- 

* For a discussion of the problem of the connecting-rod see Hill, Min, Proc. 
Inst. C. E., Vol. CXXIV, p. 390. Also consult Unwin, Min. Proc. Inst. C. E., 
CXXV, p. 363, and a paper by G. A. Burls, Min. Proc. Inst. C. E., Vol. CXXXI, 
P- 338. 



102 KINEMATICS OF MACHINES, 

do 

pin, instead of r— , and, omitting the negative sign, which 

simply shows that x diminishes at first while increases, we 
have very approximately for the velocity of the piston or 
cross-head 

F,= F.(sin.+ 5i^) (.) 

As an example, suppose an engine 12 inches stroke running 
at 250 revolutions per minute, the length of connecting-rod 

being 3 feet. The crank-pin velocity will be ~ — . ■ = 

00 

13.08 feet per second. When <9 = 45°, the value of n being 

6, we have, from equation (2), 

y^ = 13.08(0.7071 1 + 0.08333) 
= 13.08X0.79044 
= 10.340 feet per second. 

If the velocity were calculated from the accurate expression 
previously obtained, we should get 

Vp = 13.08(0.7071 1 + - —- ^ ) 

\ 2V36 — 0.4998/ 

= 13.08X0.79103 

= 10.348 feet per second. 

The approximation, therefore, has led to an error of only 
0.008 foot per second in this case. 

Proceeding to determine the acceleration of the piston or 
cross-head for any crank angle, we find very approximately 
from equation (2), remembering that V^ is constant, 

^^(F,) = Wcos^^- + _ 2 cos 2^ 



dt 



Now -- = — ^ ; thus 
at r 



V:/ . . cos 2^^ 



acceleration of piston or cross-head = —''(cos d-\-' ). (2) 

r \ n I 



SLIDER-CRANK CHAINS. 



103 



2d 
The following table gives the value of cos + cos — for 



n 



different values of 6 and n. 





Value of n. 


0. 


4 


4.5 


5 ~ 


5.5 


6 


00 


0° or 360° 


1.250 


1.222 


1.200 


1.182 


1.167 


1.000 


30° or 330° 


0.991 


0.977 


0.966 


0.957 


0.949 


0.866 


60° or 300° 


0.375 


0.389 


0.400 


0.409 


0.417 


0.500 


90° or 270° 


-0.250 


-0.222 


-0.200 


-0.182 


-0.167 


0.000 


120° or 240° 


-0.625 


-0.611 


-0.600 


-0.591 


-0.583 


-0.500 


150° or 210° 


-0.741 


-0.755 


-0.766 


-0.775 


-0.783 


-0.866 


180° 


-0.750 


-0.778 


-0.800 


-0.818 


-0.833 


-1.000 






Vnli 


. / . , cos 2e\ 
les of 1 cos B A 1 


J 














\ n 


/ 





36. Graphic Methods for Cross-head Velocity and Accel- 
eration. — We proceed to consider graphic means of deter- 
mining velocity and acceleration for the cross-head or piston 
of a direct-acting engine. It is of course possible to draw 
first a curve of displacement on a time base, and then use 
the methods described in Chapter II, but simpler means 
can be employed in this case. In Fig. 6^ let AB, BC repre- 




FlG. 63. 

sent the connecting-rod and crank in any given position. 
The point A is moving along the straight line AC, while B 
is moving for the instant in a direction perpendicular to BC. 
Hence J9, the virtual centre oi AB with regard to the fixed 



I04 KINEMATICS OF MACHINES. 

link, is easily found at the intersection of the virtual radii 
of the points A and B. Through C draw a line perpendicu- 
lar to AC, and therefore parallel to AD, and produce AB 
to meet it in £. Then the triangles ADB, ECB are similar. 
Now the angular velocity oi AB about D is measured 

either by the ratio -r-^ or by — -^ , so that 
^ AD / BD 

V/, AD _ CE 



V. BD CB 



In many positions of the mechanism D is inaccessible, 
but E can always be found, and the relation just obtained 
tells us that CE represents the velocity of the piston at the 
instant for which the diagram is drawn, to the same scale 
as that to which CB represents the velocity of the crank-pin. 

It is generally most convenient to make a polar diagram 
of piston velocity by marking off a series of points such as 
E' (where CE' = CE) for a number of different crank posi- 
tions, or, if required, a velocity diagram on a distance base 
may be constructed by marking off the distance CE along 
AD, so that a series of points such as E^' are obtained, and 
a curve drawn whose ordinate at any point is proportional 
to the velocity of the piston when in that position. Such 
diagrams have been drawn in Fig. 64, together with a linear 
velocity diagram on a time base, so as to show the difference 
between a simple harmonic motion and that which the piston 
actually possesses. The example taken is that for which the 
velocity and acceleration have been calculated in the last 
section. In order to determine the scale to which the ordi- 
nates of the curves represent the velocity, it is only neces- 
sary to remember that if the length BC were i inch, the 
velocity scale would be i inch = 13.08 feet per second, since 
the crank-pin velocity is 13.08 feet per second. In the 
figure the construction lines are shown for one position of 
the mechanism only ; in drawing such diagrams care should 



SLIDER-CRANK CHAINS. 



105 



be taken only to draw those portions of the construction 
lines which are absolutely necessary, so as to avoid useless 
complication. Of course accuracy in drawing is indispen- 
sable if the numerical results obtained are to be reliable. A 
line whose length is proportional to the piston acceleration 



motioniwere 
Harmonic 



PISTON VELOCITY ON 
DISTANCE BASE 



STROKE OF ENGINE 12 INCHES 

CONNECTING ROD CENTRES 3 FEET 
REVOLUTIONS PER MIN. 250 



360 CRANK ANGLE 
!4 SECONDS (time) 




PISTON VELOCITY, POLAR DIAGRAM 



2 FEET 



1,0 



20 



30 



40 FEET PER SECOND 



Fig. 64. 

maybe obtained as follows (see Fig. 65) : Take any given 
position of the cross-head A , and produce A 5 to cut CE in 
E. Then, as before, 

CE^^V, 
CB V~, • 

Notice that this is true whether the path of the point A 
passes through C or not, when produced. 

The desired acceleration is the rate of change of 1/^, 
which is of course proportional to the rate at which the 
distance CE is increasing or diminishing at the instant con- 
sidered. In fact the piston acceleration may be considered 



io6 



KINEMATICS OF MACHINES. 



as being proportional to the velocity of the point E along 
CE at any instant while the engine is in motion, supposing 
BE always to be in a straight line with AB. 

Let this velocity along CE be u^. The real velocity of 
the point E, regarded as a point on the connecting-rod, is 
in a direction perpendicular to DE, its virtual radius. Call- 
ing this velocity u^, we see that u^may be resolved into two 




Fig. 65. 

components, namely, u^ in a direction along CE, and U2 
in a direction along BE. 

From C draw CF parallel to DE, and draw EG perpen- 
dicular to AB. Then the sides EC, CG, GE, of the triangle 
FCG are respectively perpendicular to the directions of u^,Uq, 



u 



2' 



Thus FCG is a triangle of velocities and — 



^0-^.^^. 



u^ CG 

u^ EC 

EC 



or 



CG -u . U^ DE EC ,, r jr 

^""^V^DB^CB' therefore «, = F,^^, 



and 



T/ ^G 



= rate of change of length CE. 



Now it has been shown that 



piston velocity = F^ = CE . 



CB ' 



SLIDER-CRANK CHAINS. 107 

and V ^ and CB are constant ; hence it follows that the rate 
of change of the piston velocity must be equal to 

(rate of change of CE)X-~^, that is, 

piston acceleration=.„ii==S-C^- 

Co CB 

Thus to obtain the numerical value of the piston accelera- 
tion we must multiply the length of CG (measured to scale 

/V V 
in feet) by I — - \ , where V^ is the crank-pin velocity in feet 

per second and r is the crank throw, or radius of the crank- 
pin circle, in feet. 

Hence it follows that 

CG acceleration of piston 
TB^ V^ ' 

or, in other words, CG represents the piston acceleration to 
the same scale as that on which CB represents F//r, the 
radial acceleration of the crank-pin. 

When drawing such a diagram as Fig. 65 it happens that 
for many positions of the crank the point D becomes inac- 
cessible. Accordingly some other construction must be 
found to obtain the position of the point F, so that CG may 
be determined for any crank angle. 

Consider the triangles BEC and BAD. 
Evidently 

BE^BC 
BA BD' 

But ^57^ = ^^, because the triangles BDE, BCF are similar. 
BL) BE 

Therefore 

BE^BF 

BA BE' 
or BA.BF = BE\ 

Hence any construction which will make BE a mean propor- 
tional between BA and BE will determine the point F. 



io8 



KINEMATICS OF MACHINES. 



A number of such constructions have been given; ol 
these perhaps the most convenient in practice is that of 
Kisch.* 




Fig. 66. 

On AB describe a semicircle AHB. With centre B and 
radius BE cut the semicircle in H. Draw HFG perpendicu- 
lar to ABy cutting AB m F and CG in G. Join BH, HA. 

Then 

BF __BH 

BH~BA' 
But BE=BH. Hence BA.BF=^BE\ and CG represents 
the acceleration. 

The method of determining the acceleration scale of such 
a diagram may be shown by a numerical example. Fig. 67 
has been drawn for the engine for which the velocity of the 
piston has been previously calculated, taking a crank angle 
of 45°. The crank-pin velocity being 13.08 feet per second, 
and the connecting-rod being 6 cranks in length, we have for 
the acceleration of the piston at that particular crank angle 



acceleration = 



13.08= 



cos 45* 



cos 90' 



0-5 \ '^ 6 / 

_ 13.08 X 13.08X0. 7071 1 

= 242.1 feet per second per second. 



* See Zeitschrift des Vereines Deutscher Ingenieure, Dec. 13, 1890. Given 
also by Klein, Journal of Franklin Inst., Vol. CXXXII, Sept. 1891. 



SLIDER-CRANK CHAINS. 



109 



In Fig. 67 the actual length of the line CG, if the figure 
were drawn the full size of the engine, would be 0.351 foot. 
The radius of the crank-pin circle CB is 0.5 foot and repre- 




1000 FT. PER SEC. PER SEC. 



u^ = V^ 



Fig. 67. 
sents a velocity of 1 3 .08 feet per second. Hence the velocity 

scale is i foot = 26.16 feet per second, or ^^|- = ^^-^ — = 26.16. 

CB 0.5 

It has been shown that the rate at which the distance CE 

is changing is 

CB 
= 0.351 X 26.16 =9.18 feet per second. 

This, however, is not the numerical value of the acceleration 
required, for it represents the rate of change of a length CE, 
each foot of which stands for a velocity of 26.16 feet per 
second. Therefore, expressing u^ in feet per second per 
second, we have 

ziQ = 0.351 X 26.16X26.16 =240 feet per second per second, 
a result agreeing (within the limits of accuracy for a small 
scale drawing) with that just calculated. 

It is thus seen that to determine the scale to which CG 



no KINEMATICS OF MACHINES. 

represents the piston acceleration, we find, first, the piston 
velocity represented by unit length of CE (in this case 26.16 
feet per second) ; then it follows that a change of length of 
CE at the rate of one unit per second represents a change 
of piston velocity at the rate of 26.16 units per second, or 
a piston acceleration of 26. 16 units. But each unit of length 
of CG has been shown to represent a change of length of CE 
at the rate of 26.16 units per second, so that, finally, unit 
length of CG represents a piston acceleration of 26.16X26.16 
units. 

This relation may be expressed by saying that if the 
engine were drawn out full size and the linear velocity scale 
were i ioot = n feet per second, then the acceleration scale 
would be I foot = n^ feet per second per second. In this 
case, as in the case of all graphic methods of determining 
velocities and accelerations, the manner of finding the 
velocity and acceleration scales must be thoroughly under- 
stood ; if this is not done, the diagram becomes almost use- 
less, since no numerical values can be obtained from it. 

A number of other constructions for the piston acceler- 
ation in the direct-acting engine have been devised.* 

37. Angular Velocity and Acceleration of Connecting- 
rod. — To study the movement of the connecting-rod, adopt- 
ing the same notation as in § 35, we have, as before, 

sin d 

sm (f -- 



cos (p 



n 
Vn^ — sin^ d 



n 

The angular velocity of the connecting-rod is the rate of 
change of (p with regard to time, and we obtain at once 

d<p cos Q dd 

* See a paper by Prof. Elliott, Engineering, Vol. LIX, pp. 587 and 711, and 
Zeitschrift des v. Deutscher Ingenieure, Oct. 13, 1894. 



SLIDER-CRANK CHAINS. ni 

rifi 

Since -^ is the angular velocity of the crank, we have 
angular velocity of connecting-rod = — ^ 



Differentiating again to find the angular acceleration, we 
obtain 

d\p__\^^ J cos^ dd^ 

dt^ r 'do \/n^—sm^ 6 dt 

V'{ .d 



= —-.cosd^{n^-sm^ 6) ^ -sin 6(n^- sin^ 6) ^l 
r^ ( dd ) 

_ F/ sin d 11^—1 

^' V^'-sin'l'^'-sin' d' 

For ordinary values of n it is sufficiently accurate to write 
approximately 

angular acceleration = — — ^- . . . (2a) 

r' Vn'-sin'd 

Taking the same example as before, at a crank angle of 45° 

we have 

sin ^ =0.70711, n = 6, ^^ = 13. 08, 7=0.5. 



Thus Vw^ — sin^ 6 = 5.96 and cos i9 = o. 707 1 1 . Therefore 

angular velocity = -^ '' ; — - = s- n radians per second, 

^ ^ 0.5x5.96 

and, from equation (2), 

angular acceleration = — — . '^ — .0.987 

0.25 5.96 

= — 80.2 radians per second per second. 

Using equation (2a), we should obtain — 81.2 as a result. 

The simple construction of Figs. 65, 66, and 67 gives us 
the angular velocity of the connecting-rod. For 

angular velocity of crank = - y,, 
and angular velocity of connecting-rod = -ir. 



112 KINEMATICS OF MACHINES, 

angular velocity of connecting-rod = — - . -^-^. 

If BE is taken at the length it would have were the engine 
drawn out full size, BA -■=nr, and 

angular velocity = BE . —^. 

(Note that the real lengths of BE and r must be used, 
measured in feet, V^ being in feet per second.) 

In Fig. 67, for example, BE scales 0.355 foot, hence the 
angular velocity will be 

o • 3 5 5 X ^-^ = 3 • I o radians per second, 

0X0.25 

a result agreeing with the calculated value. 

As regards the angular acceleration we have seen that 
EC : CG : GF wm^-.u^: u^. 
The velocity u^ is the rate at which the length BE is chang- 
ing, and is therefore proportional to the rate of change of 
the angular velocity of the connecting-rod. Hence it may 
be shown (just as in the case of the velocity u^) that 

V ^ 
angular acceleration =EG . — V. 

In our example (Fig. 67) EG is 0.350 feet ; hence 

angular acceleration = o. 3 50 X , — — 

6 X0.125 

= 79.9 radians per second per second, 

a result agreeing closely with that previously obtained. 

Notice that when the crank angle is 90° EG becomes 

and therefore, if ^ = 



V ^ nr V ^ 

angular acceleration— " ' ~- '^ 



38. Angular Velocity of Cylinder in Oscillating Engine. 
Second Inversion of Slider-crank Chain. — The second inver- 
sion of the slider-crank chain is that in which the link b 



SLIDER-CRANK CHAINS. 



113 



(represented by the connecting-rod in a direct-acting steam- 
engine) is the fixed hnk. This mechanism is known as the 
swinging-block sHder-crank and is employed as an oscillating 
steam-engine, of which the link d becomes the piston and 
rod, while h is the framework. The link c is the cylinder 
and a is the crank, the cylinder swinging to and fro on trun- 
nions as the crank-shaft revolves. We proceed to compare 
the angular velocity of the cylinder with that of the crank, 
the latter being supposed to rotate uniformly. 

Let Fig. 68 represent this mechanism. The distance 
AB is the length of the fixed link, 
measured from the centre of the 
cylinder-trunnions to the centre 
of the crank-shaft, while BC is 
the half-stroke of the piston. 



Let ^- =n. 



Let the angle the 



crank has turned through from 
its lowest position be <9, ^ being 
the angle at which the centre 
line of the cylinder is inclined to 

AB. Then 

CD sin d 

tan c^ = ^p— p = -. 

^ DA n-cosd 

The angular velocity of the 
so that on differ- 



cylinder is -^ 

entiating 

d(p 



dt 



= cos (p 



= COS^ <p 



But cos^ (p = 



d sin 6 dd 
ddn — cos d ' dt 
n cos d—i do 
W' 
AD' 




(n — cos d)' 
AD' 



Fig. 68. 



(n — COS d) 



Thus 



d(p 
~dt 



AC AD'^DC 

n cos Q—\ dd 
n'—2n cos d-\-i' dt' 



{n — cos 6) ' + sin^ 6' 



114 



KINEMATICS OF MACHINES. 



From this we find by again differentiating 



d' 



9 



dt' 



ns>ind{n^-i) / doy 



which is the value of the angular acceleration of the cylinder 
for any crank angle d. 

Notice that since the angular velocity of the crank is 
uniform, the cylinder executes its forward and backward 
swings in unequal times. By assigning suitable propor- 
tions this particular inversion of the slider-crank chain 
may be utilized as a quick-return motion (see Fig. 72), by 




Fig. 69. 

causing the swinging link c to actuate the tool-box, say, of 
a shaping-machine, which can thus be made to perform its 
return or non-cutting stroke at a quicker rate, and in less 
time, than its forward or cutting stroke. 

The velocity ratio of cylinder and frame may readily be 
obtained graphically. The positions of the six virtual 
centres of the mechanism are shown in Fig. 69. Let oj^^j, 
represent the angular velocity of the crank with respect to 



SLIDER-CRANK CHAINS. 



115 



the frame h\ we wish to find o)^^. E (the virtual centre 
of c with regard to a) is a point common to the two bodies 
a and c for the instant considered. Its Hnear velocity may 
be expressed either as co^^xAE or as co^^xEB. Hence 

EB 



CO 



cb 



CO 



AE 



Draw BF parallel to CE, and EG parallel to CB. Then 

EB _CE_BG^ 
"AE~AC~BA' 



CO 



cb 



CO 



ab 







5 FEET 



5 RADIANS PER SEC. 



Fig. 70. 

and since BA is constant, the length BG is proportional to 
the angular velocity of the cylinder. A polar diagram may 



Ii6 KINEMATICS OF MACHINES. 

be drawn by marking a distance BG' along BC equal to BG 
and repeating the construction. This has been done in Fig. 
70 for an oscillating engine in which the stroke is 3 feet and 
the distance A 5 is 5 feet. At 60 revolutions per minute the 
maximum angular velocity of the cylinder is represented on 
the diagram by a distance measuring 2.14 feet to scale, hence 
its numerical value (the angular velocity of the crank being 
27r radians per second) will be 

214. 
6.28X^—^ = 2.689 radians per second. 

The angular acceleration may easily be obtained by con- 
struction from the velocity diagram, as is shown in Fig. 71. 
The value of the velocity and acceleration should be cal- 
culated for one or two positions of the crank, as an exercise, 
and compared with the diagrams. 

Notice that the value of the angular- velocity ratio when 
^=o°or 180° is 

CO c 



CO 



ab 



r(n±i) V n±i 



Notice also that the angular velocity and angular acceler- 
ation will be the same for the piston and rod as for the cylin- 
der. It is easy to show that — ^ = _- = -^_ (Fig. 69) . 

^ab ^^ ^^ 

Fig. 71 shows the angular velocity and acceleration 
(in the same example) as plotted on a polar diagram (the 
acceleration curve being found from the velocity curve as 
in § 22), and also shows the corresponding linear diagrams 
on a time base, the scales being marked. The linear dia- 
gram of acceleration could be obtained from the velocity- 
time curve by the method of § 19, but the acceleration scale 
would not then be the same as that shown on the figure. 

A number of problems dealing with velocity ratios and 
accelerations in the oscillating engine have been worked 



SLIDER-CRANK CHAINS. 



117 



out by Professor Elliott in a communication to Engineering, 
Vol. LXIII, page 665, to which the reader is referred. 



CRANK VELOCITY 6. 28 RADIANS PER SEC. 




TIME 

VELOCITY 

ACCELERATION 
DISTANCE 



0.1 0.2 0.3 04 0.5 seconds 



<L 



i_ 



_^ RADIANS PER SEC. 



5 10 

I_j i_j i_i i_i i_i I 



1 



20 RADIANS PER SEC. PER SEC 
^ FEET 



Fig. 71. 



A general method will be given later (Chapter V) by 
which the linear or angular velocity or acceleration may be 
found graphically for any point on a link of a mechanism 
of the kind discussed in this chapter. 

A good example of the kinematic identity of mechanisms 



ii8 



KINEMATICS OF MACHINES. 



which at first sight appear to be very different is afforded in 
Fig. 72. The links which correspond in the two cases have 
the same letters attached. The sketch (a) represents the 
oscillating engine, while {h) gives a diagrammatic view of 
the corresponding quick-return motion. Both are derived 




Fig. 72. 

from the same inversion of the slider-crank chain. The 
swinging link c has the same relative motions in regard to 
the links h and d (with which it pairs) in the quick-return 
motion as in the engine. The framing of the engine cor- 
responds to the fixed framework of the machine-tool. The 
rod Ry which of course does not appear in the engine, com- 
municates the variable motion of the swinging link c to the 
tool -carriage. The crank-shaft o^ the engine is represented 
by the disc a, to which rotary motion is imparted by the 
driving mechanism of the tool (not show^n). 

39. Whitworth Quick-return Motion. Third Inversion of 
Slider-crank Chain. — Passing to the next inversion, we 
now have a as the fixed link, and the resulting mechanism 



SLIDER-CRANK CHAINS. 



119 



is one which was appHed by Whitworth* as a quick-return 
motion for the same purposes as have already been men- 
tioned. It has been called by Reuleaux the turning-block 
slider-crank chain. 

The velocity ratio of the links h and d may be obtained 

Fig. 73a. 





Fig. 73(5. 

exactly as in the case of the oscillating engine. In fact an 
alteration in the relative lengths of the links a and 6, Fig. 70, 
converts the mechanism there shown into one not differing 
in any essential particular from the quick-return motion 
of Fig. 736. 

* See Cotterill, Applied Mechanics, p. io8, § 50. 



I20 KINEMATICS OF MACHINES. 

Figs. 73a and 736 show this quick-return motion with 
different proportions of the lengths of the hnks a and h. It 
will be seen that while d executes complete rotations in the. 
first case, it only swings in the second case. 

The relative angular velocities of h and d are easily found 
by the construction shown in Fig. 69. Evidently if a is the 
fixed link 

CO, BD BA GA 

da 



oj, CD EA BA' 

ba 



Thus a AB represents the angular velocity of the link b^ 
AG represents that of d to the same scale. 

Supposing the link b to have a uniform angular velocity,, 
the positions i and 2 are those in which the tool-box is at 
one or the other end of its travel. Accordingly it is easily 
seen that the times of the cutting and return strokes will 

be in the ratio . Hence in designing such a motion 



a 

a 



we have only to proportion a and b so that t = cos a in thc: 

Whitworth motion, or - =- cos a in the other form, where 

a 

a has such a value as to make the desired ratio. 

a 
The times of the cutting and return strokes are often in the 
ratio 2 : I or 3 : I. 

The centrodes for the links c and a are found by similar 
constructions to those already shown in Fig. 61 for the links 
b and d, and are drawn in Fig. 74. The reader should con- 
struct them for himself for the case shown (in which the 
length of the link a is less than that ofb), and also for the case 
in which the link a is longer than b, when the centrodes take 
quite different forms. 

40. Pendulum Pump. Fourth Inversion of Slider-crank 
Chain. — The last of the four possible inversions of the chain, 
the swinging slider-crank, in which c is the fixed link, has. 



SLIDER-CRANK CHAINS, 



121 



only a very limited application in practice, but has been 
employed as a small steam donkey -pump. It is shown 
diagrammatically in Figs. 75a and 756, and is shown also 
in outline in the second sketch in Fig. 74. 




\. 




a FIXED C FIXED 

Fig. 74. 
It is known as the "pendulum pump," from tHe motion 
of the link h. The link c now represents the cylinders 
(steam and water) and their connecting framework, while 
d is the piston, rod, and plunger. The crank a takes the 
form of a small fly-wheel, which rotates about 0^^, while that 
point swings along the dotted arc. The relative angular 
velocity of h and d is easily found graphically, the virtual 
centres being known. Let F = linear velocity of d\ then, 
since the link a is turning for the instant about E, we have 



angular velocity of a = 



linear velocity of point B _ V 



BE 



CE' 



and 



linear velocity of point B = V . 



BE 
CE 



122 



KINEMATICS OF MACHINES. 



Therefore 



angular velocity oih = V 



= F 



BE 



BAXCE 
BA 



V 



BA XAD AD • 





pD i-Obd) 



-bE (rOac) 



Fig. 75a. Fig. 75^. 

The various inversions of the slider-crank chain may be 
summarized thus : 

Link Fixed. Name of Chain. Resulting Mechanism. 

d Turning slider-crank Direct-acting engine 
b Swinging-block slider- Oscillating engine. Quick- 
crank return motion 
Turning-block slider- Whitworth quick-return mo- 
crank tion 
Swinging slider-crank Pendulum pump 
Crossed Slider-crank Chains. — The slider-crank chains 
hitherto discussed have been arranged so that the straight 
line in which 0^^ moves relatively to d passes through 
O ^. If this is not the case, we obtain a further series of 

aa ' 

mechanisms known as crossed slider-crank chains, shown 
in Fig. 76. 



a 

c 
41. 



SLIDER-CRANK CHAINS. 



123 



. The crossed turning slider- crank has been used in certain 
single-acting high-speed steam-engines, with a view of lessen- 
ing the effect of the obliquity of a short connecting-rod 
during the working stroke ; the obliquity during the return 
stroke is of course correspondingly increased. In such a case 
the determination of the acceleration or velocity of the 





CROSSED TURNING S.C. 



//////M 

CROSSED SWINGING BLOCK S.C. 




CROSSED TURNING BLOCK S.C. 




CROSSED SWINGING S.C 



Fig. 76. 

piston does not present any difficulty, as it can be carried 
out by the constructions already given. 

42. Double Slider-crank Chain. — We consider next the 
simple chain formed by two turning and two sliding pairs. It 
has been already shown that from the quadric crank-chain the 
slider-crank chain may be derived by substituting a sliding 
pair for one of the turning pairs, such sliding pair being 
equivalent to a turning pair of infinitely great radius. This 
substitution may be repeated, and Fig. 7 7 shows the result 
in the case where the directions of motion of the two sliding 
pairs are at right angles, and where one link carries an ele- 
ment of each of the two sliding pairs. Such a chain is called 
a double slider -crank chain. The link h has now become 
compressed into a block sliding in a slot formed -in c. 



124 



KINEMATICS OF MACHINES. 



Since the relative motion of h and c is the same as if the 
pair be were a turning pair of infinite radius, the velocity 
ratios and accelerations in this chain will all be found exactly 
as in the case of a slider-crank chain in which the link h is of 



-e — 



^ 






^. 



a 



■\ 



\ 



Fig. 77. 



1 




Fig. 78. 

infinite length, i.e. when ^ = 00 . Accordingly we may 
write at once (supposing the link d to be fixed), with the 
same notation as before, 

x=-Y cos Q, 
and 

linear velocity oic^Vp^V ^ sin ^, 



SLIDER-CRANK CHAINS. 



while 



125 



acceleration of ^ = 



V. 



cos d. 



Notice that if the crank rotates uniformly the motion of 
c with regard to d is simple harmonic, and that the link h 
has no angular velocity with regard to d. 

Fig. 78 shows the six virtual centres of the chain. It is 
plain that the centrodes of h and d are now altogether in- 
accessible. 

The distance CE is seen to be proportional to the linear 
velocity of the link c, while CF is proportional to its linear 
acceleration as given above; the scales are readily deter- 
mined. 

This inversion of the double slider-crank chain is fre- 
quently employed in the construction of steam-pumps. The 
link c represents the steam-piston 
and pump-plunger, d the cylinder, 
framing, and pump-barrel, and a 
the crank-shaft. The total height 
of the pump may be made small, on 
account of the absence of a con- 
necting-rod, thus making the ar- 
rangement a very convenient and 
compact one for certain purposes. 

The linear and polar diagrams 
of piston displacement, velocity, 
and acceleration, supposing that 
the angular velocity of the crank- 
shaft is uniform, are precisely those 
already given for simple harmonic 
motion. 

If the link h be supposed fixed 
instead of d, the resulting mechan- 
ism is the same as before, for the 
reason that the relative motion of 
h and a is exactly the same as ^°' ^' 

that of d and a. Thus on fixing h we still have the link c 




126 



KINEMy^TICS OF MACHINES. 



executing simple harmonic vibrations as the crank rotates 
uniformly, but the direction of motion is now along the line 
FB instead of FC (Fig. 78). 

43. Elliptic Trammels. — If c, the link containing one 
element of each of the two sliding pairs, is the fixed link, 
we obtain a mechanism used for the purpose of drawing 
ellipses, and shown in Eig. 80. The bar a, carrying an ele- 
ment of each of the two turning pairs, now carries a mov- 
able tracing-point ; the blocks b and d slide in a pair of grooves 
intersecting at right angles and formed in the link c. 




Fig. 80. 



The path of the tracing-point is easily seen to be an 
ellipse, for, with the notation of Fig. 80, we have 

sin d =y/n, 
cos d = x/m. 



Hence 






This equation is seen to represent an ellipse having as 
its centre and m and n as its major and minor semi-axes. 

From the position of the point 0^^ (Fig. 81) it is evident 
that the centrodes of a relatively to c, and c relatively to a. 



SLIDER-CRANK CHAINS. 



127 



form a pair of circles of which the length of the link a is 
respectively the radius and the diameter. Hence it follows 
that the relative motion of a and c may be represented by 
the rolling together of circular curves of the same sizes as 
the centrodes in question — a point to which attention is 
again drawn (see § 57). 




Fig. 81. 
Notice that if V ^, V^ be the linear velocities of d and b 

respectively with regard to c, then, since oj^^ = -^ = ^ykr, 



OB 



we have 



L 

F. 



OB 
OD' 



44. Oldham's Coupling. — The fourth inversion of the 
double slider-crank chain, when a is the fixed link, gives 
rise to a mechanism which has been used as a coupling 
for connecting shafts whose axes are parallel, and as an 
elliptic chuck, by means of which objects of elliptical cross- 
sections may be turned in an ordinary lathe. 

Let Fig. 82 represent the chain when a is fixed. Evi- 
dently the locus of is a circle of diameter BD. 

Let Oj, O2 be two positions of ; then, since OJDO^ and. 



128 



KINEMATICS OF MACHINES. 




Fig. 82. 




Fig. 83. 



SLIDER-CRANK CHAINS. 



129 



O^BO^ are angles in the same segment of a circle, they are 
equal; hence if h turns through any angle O^BO^, d turns 
through an equal angle OJDO^. By attaching a shaft to 
each of the links h and d we are thus enabled to commimi- 
cate rotation from one to the other with uniform angular 
velocity ratio. Fig. ^2> shows the form actually taken by 




Fig. 84a. 



this mechanism when used as Oldham's coupling. The link 
c becomes a disc having projecting feathers or keys on its 
faces, these keys being at right angles to one another and 
fitting into corresponding grooves on the enlarged ends of 
the two shafts h and d. The link a becomes a frame carrying 
the bearings of the two shafts. 



13© 



KINEMATICS OF MACHINES. 



Precisely the same kinematic chain is used in the case of 
the elliptic chuck, which was probably invented by Leonardo 
da Vinci. 

Figs. 84a and 846 represent this contrivance, seen from, 
the back, the face of the plate c to which the work is 
attached being turned away from view. 

The plate c has behind it two straight pairs of guides 
at right angles to one another; the block h slides be- 




FiG. 84/^. 

tween one of these pairs of guides, while the block d moves 
between the other pair, which pass through slots cut for the 
purpose in h. The block h is secured rigidly by being screwed 
on to the nose of the lathe mandrel, with which it rotates. 
The mandrel passes through an oval hole in the eccentric a, 
which is clamped firmly (by screws which are not shown) to 



SLIDER-CRANK CHAINS. 



131 



the lathe neadstock, in such a way that the distance BD be- 
tween the axis of a and that of h can be varied as required. 
That distance is the effective length of the fixed link in the 
mechanism, and upon it depends the eccentricity of the 
ellipse to be described. 

It will be seen that this construction corresponds ex- 
actly to the arrangement of Fig. 82. Accordingly it is 
evident that a point at rest with regard to the link a (as the 
point of a cutting-tool would be) will describe an ellipse on 
a piece of work attached to, and rotating with, the link c, 
just as a tracing-point attached to a (Fig. 80) was shown 
to describe an ellipse with respect to c in that case. It 
will be seen that the distance from the tracing-point P to 
D (Fig. 846) is the semi-minor axis, while the length BD 
is the difference between the semi-axes. 

It is obvious that a number of fresh mechanisms may 
be derived by changing the angle between the directions of 
motion of the two sliding pairs ; in this case the chain would 
be known as a skew double slider-crank chain. Fig. 85 
shows an example of such a chain, but space does not per- 
mit of the discussion of such mechanisms. 




Fig. 85. 



45. Crosse d-slide Chains. — We proceed to consider the 
chain derived from the slider-crank chain by introducing 
a second sliding pair in such a way that each link contains 
one element of a sliding and one of a turning pair. As 



132 



KINEMATICS OF MACHINES. 



distinguished from the chain just discussed (the double 
slider-crank chain), in which one link contains elements of 
each of two turning pairs, and another contains elements of 
each of two sliding pairs, we may call this the crossed-slide 
chain. It is essentially a crossed chain, just as the crossed 
slider-crank was, because the straight line in which the centre 
of one turning pair moves does not pass through the centre 
of the second turning pair. One of its forms is shown dia- 
grammatically in Fig. 86, and is occasionally employed for 




Fig. 86. 



working the rudder of large ships, under the name of 
Rapson's slide. For this purpose it has the great advantage 
that the leverage increases as the helm is put over. In the 
figure the fixed link d represents the framework of the ship, 
a the tiller and rudder-head, and ^ is a block sliding on a 
and turning on c. The steering-gear moves the block c 
between guides secured to d, and thus actuates the rudder. 

The same mechanism is employed for working the valves 
of duplex steam-pumps, in which each of the two steam- 
pistons works the valve of the neighboring cylinder, and it 
occurs again in a slightly different form in the arrangement 
of the compensating cylinders used in the Worthington high- 
duty pump for storing up the excess of energy exerted by 



SLIDER-CRANK CHAINS. 



133 



the steam during the first portion of the stroke of the piston, 
and restoring that energy during the later part (Fig. 87). 
Here d is the pump framework, a a are the compensating 
cyhnders, rocking on trunnions attached to d; b b are the 
plungers which enter the cyHnders against pressure during 
the first half of the stroke, and return during the later half ; 
c is the pump-piston, rod, and plunger. 

The virtual centres of the chain are shown in Fig. 88, and 




Fig. 87. 

the pair of centrodes corresponding to the relative motion 
of 6 and d are shown in Fig. 89, construction lines being 
given for one point on each centrode. 

Certain velocity ratios in this chain are of importance; 
for example, the ratio of the angular velocity of the tiller a 
to the linear velocity (FJ of the block c relatively to its 
guides. 



A(=Oad) 




.--^ 



}- — 



Ocd 

AT INF. 

Fig. 88 



In Fig. 88 let the angle BAG = 6 ; then, since 0^^ is a point 
common for the instant to a and c, as a point in c it is moving 



134 



KINEMATICS OF MACHINES, 



in a direction perpendicular to AC with velocity V ^. Its 

angular velocity (and therefore the angular velocity of a 

V 
in which it is a point) about A is therefore ^-^, which is 

easily seen to be equal to 

V 

-ir-^ cos 2 d. 
AE 

Hence if the block c has a uniform linear velocity, the angular 
velocity of the tiller varies as the square of the cosine of the 




Fig. 89. 

angle of helm. It is this property which gives the apparatus 
its value as a steering-gear ; for it may readily be shown that 
if a constant force be applied to c, the turning moment on 
the rudder-head increases as the helm is put over; ia 

fact the tumiag moment varies as — 5-.. 
^ cos^ d 



SLIDER-CRANK CHAINS. 



135 



It is easy to draw a curve of angular velocity for the 

link a. In Fig. 90 make AE' =AE and draw BC perpen- 

AF AE' 
dicular to AB. Draw EF parallel to CE' , then ~—=-—\ 

therefore -;rT^o=^r7-' ^^^ 
AE^ AC 

angular velocity of tiller = A F.— ^2- 

Thus a series of points such as F will give us a polar 
diagram of the angular velocity of the tiller. 




Fig. 90. 

Notice that in any position we may look upon V ^, the 
linear velocity of the point B' along EB', as being the result- 
ant of two velocities, V ^ sin 6 along AB\ and V ^ cos d at 
right angles to AB\ The former gives the speed with which 
the block h is sliding along a ; the latter shows that the angu- 
lar velocity of a is 

y ^_cos^ _ V^ cos^ 6 
AB' " AE ' 

the same result as that obtained previously. 



136 



KINEMATICS OF MACHINES. 



Rapson's slide is only a particular case of the crossed- 
slide chain. It may be noticed, however, that we obtain 
the same mechanism v/hichever link is the fixed one, since 
each link has on it an element of a turning pair and also an 
element of a sliding pair. 

46. Straight-line Motions Derived form Slider-crank 
Chain. — ^A number of straight-line motions have been de- 
vised which are really slider-crank chains. In such mech- 
anisms the line described by the tracing-point is often only 
approximately straight, and when it is exactly so, its straight- 
ness depends upon the accuracy with which the flat surfaces 
of the sliding pair have been formed. 

To this class belongs Scott Russell's straight-line motion,, 
represented in Fig. 91. The link h in an ordinary slider- 




FlG. 91. 



crank chain is extended to E, and AB = BE = EC. It is then 
evident that the angle ^CE is the angle in a semicircle, and 
that the point E describes a straight line CE so long as A 
describes a straight line AC. 

With other proportions of the lengths AB, EC, EE, 
approximate straight-line motions may be obtained. In 
Fig. 92, for example, suppose A and E to lie on the straight 
lines AC, EC, respectively; it has been seen that a point B 
will describe an ellipse (shown by the dotted curve), of 
which C is the centre, and AE and BE the lengths of the 
semi-axes. A circle may be drawn so as to cut this ellipse 



SLIDER-CRANK CHAINS. 



137 



in four points, as at P, Q, R, 5, and if we connect B and F, the 
centre of the circle, by a rigid link, the path of the point E 
will cut the straight line CD in four places, supposing 
A traverses the straight line AC. By a suitable choice of 
the point F, the circular path of B may be made to differ 




Fig. 92. 

very little from the ellipse during a considerable range of 
movement, and the actual path of the tracing-point E will 
nearly coincide with the straight line CE. 

In the second inversion of the slider-crank chain, in which 
b is the fixed link, a point on the link d may be chosen such 
that its path is approximately straight. 

Thus in Fig. 93a suppose that a straight line AB, of fbced 
length, passes through a fixed point 0, while a point C on it is 
compelled to traverse a straight line DE. The curves de- 
scribed by A and B are known as conchoids, and are shown 
by the dotted lines. It is possible in a swinging-block slider- 
crank chain to find a point P on the link d in such a position 
that while the circular path of 0^^ coincides nearly with the 



138 KINEMATICS OF MACHINES. 

part BB' of the conchoid, and the centre of the pair he cor- 




FiG. 93a. 

responds with the point 0, the path of the tracing-point 
P will nearly coincide with DE. Fig. 936 represents a 
model of such a mechanism, in which the point P is guided 
in an approximately straight line.* 

47. Chain Containing Sliding Pairs only. — It is possible 
to construct a closed kinematic chain containing only slid- 
ing pairs. Such a chain consists of three links, a, h, and c. 
Of these h and c are blocks sliding on a common link a. 
A projection on h slides in a slot cut in c, thus completing the 
chain. If c is the fixed link, the chain will evidently be 

* For discussion of a number of straight- line motions see Kennedy, Mechanics 
of Machinery, p. 417. 



SLIDER-CRANK CHAINS. 



139 




Fig. 93^. 




Fig. 94. 



I 



I40 KINEMATICS OF MACHINES. 

capable of being moved into such a position as that shown 
by the dotted lines in Fig. 94. 

vSuch a chain exists, for instance, in most arrangements 
for adjusting bearings by means of wedges or cotters, as in 
the double-adjustment plummer-block (sketched diagram- 
raatically in Fig. 95), in which the 
brass c has to be capable of slight 
movement in the direction of the 
arrow, to allow for wear, and is 
pushed forward by drawing down 
the wedge h. The pedestal itself 
and its cap form the link a, and the 
upward movement of the block or 
wedge h is prevented by some form 
of force- or chain-closure (see Chapter VI). 

A chain containing four links and having four sliding 
pairs can also be devised, but, like the chain containing five 
turning pairs, it is not constrainedly closed. 




Fig. 95. 



CHAPTER V. 

DETERMINATION OF VELOCITY AND ACCELERATION IN 

PLANE MECHANISMS. 

48. Velocity and Acceleration Determined from Virtual 
Centres. — It is often necessary to determine the magnitude 
and direction of the velocity or acceleration of a given point 
of a given link in a plane mechanism. Such a calculation, 
for example, is frequently required if we wish to find the 
forces acting on a part of a machine when in motion, with a 
view to the correct proportioning of such a part to the work 
it has to do. 

We have already studied this problem in certain cases, 
especially as regards the cross-head of a direct-acting steam- 
engine; the question has now to be discussed in a more 
general manner. 

In a given mechanism, having given the velocity of a 
point on one link, and having found the positions of the 
various virtual centres, it is possible to determine the veloc- 
ity of any point on any one of the links. 

Take for example the beam-engine of Fig. 96, in which 
we suppose F^, the velocity of the crank-pin, to be known. 
It is required to find the actual linear velocity (i.e., the 
velocity with relation to the frame or fixed link) of the 
piston and rod b. 

Let a be the fixed link, b the piston, d the beam, e the 
connecting-rod, and / the crank. 

First find 0^^, at the intersection of a horizontal line 

through the beam centre 0^^ and the line joining 0^^ and 

^. Note that 0^^ is at an infinite distance. Next find 

141 



142 



KINEMATICS OF MACHINES, 



Oaf, draw a horizontal line through O^/* and find its inter- 
section with the line joining O^^ and O^y. This point is Oj,f. 




Fig. 96. 
All these centres are readily found, remembering that they 
lie in threes in straight lines. 

The point O^y is a point common for the instant to the 
links h and /. Let the length of crank = r and let the dis- 
tance Oaf . . . Oi,f=m, 

Then the actual linear velocity of the point O^y (consid- 



m 



ered as a point on the link /) must be F . X — , in a direction 

r 

perpendicular to the line OafOi,^, and this must also be the 

velocity of the link h, since O^y is for the instant a point on 

that link also. The same construction will give the 

velocity of the piston for any position of the mechanism, 

except when the crank is on the dead-centre. 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 143 

A similar method may be used in any case in which the 
various virtual centres can be found, but is not always possi- 
ble for all positions of the mechanism, because many of the 
centres periodically recede to an infinite distance. This 
fact considerably reduces its practical usefulness. 

49. Method by Using Point-paths. — The velocity of a 
given point on any link may be most simply determined for 
any given position of a mechanism by carefully drawing (to 
as large a scale as possible) the mechanism in two positions, 
one slightly before and the other slightly after the given 
position. The velocity of some one point of the mechan- 
ism being known, the velocity of the given point is readily 
found by comparing the displacements of the two points 
in the short time supposed to elapse between the two posi- 
tions drawn, the direction of motion being known from 
the point-path on the drawing. It should be noted that 
this method of finding the velocity required is not suscep- 
tible of great accuracy, because the displacements whose 
ratio is measured must be supposed very small, in order 
that the result obtained may be as nearly as possible the true 
velocity of the point when the mechanism is actually in the 
given position. Hence the ratio of the displacements is 
difficult to measure. The method is nevertheless often 
used in practice. 

As an example the mechanism of Fig. 97 may be taken. 
The figure shows Bremme's valve-gear.* It consists essen- 
tially of a lever-crank chain, the motion of the valve being 
taken from a point on one link produced. The figure, neces- 
sarily drawn here to a small scale, shows the proportions of 
an actual gear for a small marine engine. The eccentric 
of the engine corresponds to the crank of the lever-crank 
chain, and in practice coincides in angular position on the 
shaft with the engine-crank. The dimensions are: 

* See Mechanical World, September 2, 1889. 



144 KINEMATICS OF MACHINES. 

AC == if" (throw of eccentric) ; 
CE = ioV'\ 

EB = 14''; 

AB = 20" (when the engine is going ahead). 
The engine is reversed by altering the position of the suspen- 
sion point B, as shown by the dotted arc. 

It is required to determine the vertical component of 
the velocity of the point D (from which the valve is driven 
by a long rod) for any position of the gear, supposing the 
eccentric AC rotates uniformly at a speed of 170 revolutions 
per minute. 

We first take a number of positions oi AC (in this case 
1 2 in one revolution) , corresponding to equal small intervals 
of time (in this case 0.0294 second), and the corresponding 
positions of E and D are found. They are shown on the 
diagram and numbered successively, those of D forming 
points on a closed curve roughly oval in shape. 

The vertical displacements of the point D have been 
plotted on a time base, giving the diagram xx. From this 
the velocity curve YYY has been drawn by the method of 
§18. On determining the scale of the diagram we see that 
between the positions 4 and 5 the valve moved 0.60 inch 
upwards in J^- revolution, i.e. in 0.0294 second. The veloc- 
ity of the valve at the middle of that interval will therefore 
be approximately 

'- =1.70 foot per second. 

12 X0.0294 

In the same way the maximum downward velocity of the 

valve is found to be while the crank is moving from 10 to 11, 

and its value is about 2 .00 feet per second. 

If necessary the vertical acceleration of the valve can be 
determined as in § 19. 

On drawing out the example for himself the reader will 
find that even if the mechanism be drawn full size great 
care is necessary to obtain anything like an accurate result. 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. i45 



CENTRE LINE 







146 



KINEMATICS OF MACHINES. 



50. Polar Diagrams of Velocities for Simple Plane Mech- 
anisms. — The velocities in plane mechanisms can only be 
determined graphically from the positions of the virtual 
centres of the links when these centres fall within the limits 
of the drawing, and when their positions can be found with 
accuracy. Often the exact position of a virtual centre is 
difficult to define, because it lies at the intersection of two 
lines which make a very small angle with one another. 

To avoid these difficulties, a general method of drawing 
diagrams for velocities and accelerations of points in mech- 
anisms has been devised,* and a few simple cases will be 
considered here. 

In Fig. 98 let ABC represent a rigid body having plane 

motion, and suppose the linear 
velocity v of the point A and 
the angular velocity co of the 
whole body to be known. It is 
required to determine the linear 
velocities of the points B and C. 
Let P be the virtual centre 
of ABC with regard to the plane 

of motion ; then AP = -- hence 

w' 

the position of P can be found, 
since PA is perpendicular to the 
direction of v. 

Join PB, PC. From these 
lines the directions of motion of 
B and C are known, and the 
magnitudes of the linear veloci- 
ties are also known, since 
velocity oiB = o}X PB, 





Fig. 98. 



and 



velocity ofC^coX PC 



* R. H. Smith, Graphics, Book I, Chap. IX ; Burmester, Kinematik, Chaps. 
XI and XII. 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 147 

These velocities can, however, be determined (without 
finding the position of the virtual centre) as follows : 

From any pole ^draw the vector pa, representing the 
velocity v. From the point a draw ah{ = (xjXAB) perpen- 
dicular to AB, and draw he ( = coXBC) perpendicular to BC. 
Then ph, pc are vectors representing respectively the linear 
velocities of the points B and C. The truth of this state- 
ment will be seen from the facts that the sides pa, ah are 
perpendicular to the sides PA, AB, and they are also pro- 
portional, since 

pa = ioXPA, 
and ah = ojxAB. 

Hence ph = ojX PB = velocity of B. 

And similarly pc = coXPC = velocity of C. 

Note also that the triangles ahc, ABC are similar; in 
fact ahc is the velocity image of the body ABC, and is turned 
through an angle of 90° in the same sense as that of the angular 
velocity co. The lines ah, he, ca are of course vectors, and on 
consideration it will be evident that ah, for instance, 
represents the linear velocity of B (round A as centre), 
due to the actual angular velocity oj, because we have 
drawn ah = oj.AB and at right angles to AB. Further, 
the values of ph and pc have been obtained by vector- 
addition, the process described and explained in § 16. 

Next suppose that we have to determine the veloci- 
ties in a linkv^^ork mechanism such as that of Fig. gg, 
where ZPABC represents a chain of links connected by 
turning pairs, PZ being the fixed link. 

The angular velocity co of PA is supposed to be known, 
and also the directions in which the points B and C are 
moving at the instant considered 

The point A is turning about the fixed point P ; hence its 
direction of motion is at right angles to PA and its linear 
velocity is oj . PA = v^. We have to find v^ and v^, the linear 
velocities of B and C. 

Take any pole p, and draw lines pa, ph, pc from it par- 



148 



KINEMATICS OF MACHINES, 




Fig. 99. 



allel to the given directions of v^, v^, and Vc. Set off pa =v^ 
and draw ab perpendicular to AB and be perpendicular to 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 149 

BC. Then pb and pc represent v^ and v^ on the same scale 
as that on which pa represents v^. 

The triangle pab, for example, is a vector triangle, or 
triangle of velocities, for the body AB, and ab is the velocity 
image of AB, just as in the previous case. The vector ab 
really represents the velocity of A with regard to B or of 
B with regard to A, according to the sense in which we 
measure it. 

It is evident that the linear velocity of the point B will 
be due to two causes: (i) the velocity of A with regard to 
the fixed link: ZP, and (2) that of B with regard to A. We 
also know that B can have no velocity along BA , for BA is 
a rigid body. , 

Hence to find pb (the velocity of the point B with 
regard to ZP) we compound pa (the velocity of A with 
regard to ZP) with ab (the velocity of B with regard to 
A). Similar reasoning holds good in the case of pc. 

If cb be produced and a line, pn, drawn to cut it at right 
angles, it will be seen that pc is the resultant of pn (the 
velocity of C in the direction CB) and nc (the velocity of C 
in the direction normal to CB) . Similarly nb is the velocity 
of B in the direction normal to CB, and hence be is the veloc- 
ity of C with regard to B, measured in a direction normal to 
CB, i.e., be is the velocity of C about B. Note that since 
the link CB is rigid, C and B must have the same velocity, 
pn, along CB. 

The diagram can be drawn in a similar way if some of the 
pairs are sliding pairs, and it will be found that if the chain 
of links has both ends attached to fixed points the velocity 
diagram becomes a closed polygon. 

If it is required to find the velocity of a point D in the 
link BC, that of C being known, it is only necessary to 

TDTJ 

make bd = beX-^^y and to join pd. The required velocity 
is then represented by pd. This is evident, since the veloc 



ISO 



KINEMATICS OF MACHINES. 






e^\. 



t?M8 

1 1 — 
If 









/ 



,1^10 



/ 



/ 

V 



m^ 



\ 



\ 



w 



N 



10 





FEET PER 
3 SECOND 



Fig. too. 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 151 

ity of D about B is to that of C in the proportion of BD : BC, 
hence r-= dT^- Thus bed is seen to be the velocity image 

of BCD. 

We may take as an example of the use of this construc- 
tion the Bremme valve-gear of Fig. 97. In Fig. 100 the 
velocity diagrams have been drawn for the positions 4, 8, 
and 10 of Fig. 97. The diagrams have been drawn from 
separate poles for the sake of clearness, but they might 
equally well have been drawn from the same point as pole 
if that had been advisable. 

Having drawn the mechanism in position 4 (say), and 
having found by calculation that v^ (the velocity of the 
centre of the eccentric) is 2.59 feet per second, a line p^c^ is 
•drawn from the pole in a direction parallel to i'^, and of the 
proper length. We know that the direction in which E 
is moving is perpendicular to BE, and p^e^ is therefore drawn 
of indefinite length perpendicular to BE. The point e^ is 
found by drawing c^e^ perpendicular to C^E^ and p^e^ then 
gives the magnitude of the velocity of E. To find that of 

D, we produce c/^ to d^, making — = -^ ; then p^d^ represents 

the velocity of D when the mechanism is in position 4. The 
vertical velocity of the valve will be represented by d^m^, 
the vertical velocity of the point D to which it is attached, 
and on measurement this line is found to scale 1.56 feet per 
second. (Compare value shown by curve in Fig. 97.) In 
a similar way d^m^ and d^^m^Q are found, and so on for any 
required position of the gear. 

It is not difficult to see that the velocity diagrams ob- 
tained by this method are really the same as some of those 
whose construction in certain special cases is explained in 
Chapters III and IV. For instance. Fig. loi shows the 
construction already described for the piston velocity in a 
■direct-acting engine, together with the polar method of 



152 



KINEMATICS OF MACHINES. 



determining the same quantity. It is plain that the tri- 
angles BCE and hpa are similar and that pa and CE repre- 
sent the same quantity to different scales. The vector pa 
thus represents the linear velocity of the piston, while ph 




\ 


V 






/ E 


^\b 






4 




^ ^ 


A 


\ 


y 




^ — 



Fig. ioi. 



and ah represent respectively the linear velocity of the crank- 
pin B around C and that of the crank -pin B around A. ah 
is in fact the velocity image of AB, and pd gives the velocity 



of any point D on AB, li — == — — ' 



Since the triangles BCE 



and hpa are similar it also follows that we may look on BE 
as a velocity image of BA. It is, however, turned through 
an angle of 90° from the position ah. 



VELOaTY AND ACCELERATION IN PLANE MECHANISMS. -53 

,1 Indirect Method in more Complex Cases.-It is not 
possible in every case to proceed in such a direct manner m 




ronstructing the velocity diagram for a mechanism. Fig 
rshowfa link motion for working the slide-valve of a 



154 KINEMATICS OF MACHINES. 

steam-engine, of which OA, OB are the two eccentrics. 
These are rigidly connected and rotate uniformly about 0, 
the centre of the crank-shaft ; AC, BF are the eccentric -rods, 
CF the link, and QE is the drag-rod or suspension-link. 
Q remains fixed, except when the engine is reversed. • We 
wish to find the velocity of a point D on the link CF. 
The motion for the valve is taken from the point D. 

The actual velocities of points A and B are known, and 
also the direction of motion of E. 

Having drawn out the mechanism in the required posi- 
tion, a pole, p, is taken and the vectors pa, pb drawn repre- 
senting the linear velocities of A and B respectively. In the 
figure these correspond to an angular velocity of 120 revolu- 
tions per minute ; they are each 4. 2 feet per second. 

A line px of indefinite length is next drawn at right angles 
to QE, and therefore parallel to the direction of motion of E. 
The point e must of course lie somewhere on this line. We 
next draw ay, hz of indefinite length, perpendicular to ^C 
and BF respectively ; the points c and / must lie somewhere 
on these lines. 

The required velocity images of the points C, E, and F 
must lie in some such position as ^0^0/0* where the triangle 
^0^0/0 is similar to the triangle CEF, but is rotated through 
90° in the sense of the motion. Another possible position 
would be c^ej^, and the line e^e^ will evidently pass through 
all possible positions of e. Thus e is found at the intersec- 
tion of the lines e^e^ and px. We then draw ec and ef re- 
spectively perpendicular to EC and EF, and through the 
points c e and / draw a circular arc, which will be the 
velocity image of the curved link CEF. 

The vectors pc, pe, pf then give the velocities of C, E, 
and F respectively. To find the velocity of D the point d is 
marked in its proper position on the curve cef, and pd is 
drawn. In the position shown this velocity is 1.42 feet per 
second, and is not in the direction of the centre line of the 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 155 

valve-spindle. It is therefore necessary to attach the valve- 
spindle to a link-block capable of sliding on or in the link. 
The horizontal velocity of the point D and the actual vertical 
velocity of the valve will be represented by the lines pm and 
dm ; they are respectively 0.35 and 1.36 feet per second. 

The indirect method just explained has to be adopted 
to draw the velocity diagrams for many compound kine- 
matic chains. The exaraple here given will be sufficient to 
guide the reader in constructing such diagrams for most 
cases occurring in practice. 

52. Polar Acceleration Diagrams for Plane Mechanisms. 
Acceleration Images. — Since accelerations, like velocities, 




Velocity Diagram. 

Fig. 103 
are vector quantities, it is plain that by similar constructions 
to those explained in the preceding section we can obtain 



156 KINEMATICS OF MACHINES. 

polar acceleration diagrams ; there is one pole from which 
the vectors radiate, just as in the corresponding velocity- 
diagrams. 

Referring again to the rigid body of Fig. 98, suppose 
that we know the linear acceleration v' of the point A and 
also the angular velocity oj and angular acceleration 10' for 
the body. We wish to find the linear accelerations of 
B and C. 

In Fig. 103 take a pole, p\ and draw p'a\ represent- 
ing to any convenient scale the known acceleration of the 
point A. 

The acceleration of the point B may evidently be ob- 
tained by adding to the vector p'a' the vector a'h\ repre- 
senting the acceleration of B with regard to the point A . 

Now, the acceleration of a point moving with uniform 

linear velocity v (or angular velocity co) in a circular path is 

t;2 . 

— , or it>V radially (§ 15). But if the linear velocity be not 

r 

uniform the point will be subject not only to the radial 
acceleration just mentioned, but also to a tangential accel- 
eration directed along the path of the point and measured 

civ 
by v^ =-!-> the rate of change of the magnitude of the veloc- 

ity. This quantity may also be denoted by 

dv d doj , 

-j-=-j-r(o=^r-^ -=ruj\ 
at at at 

where r is a constant. We thus see that in order to deter- 
mine fully the acceleration of a point we must know not only 
the rate of change of direction of the velocity, or radial 
acceleration (wV), but also its rate of change of magnitude, 
or tangential acceleration {rco') , w being the angular velocity 
and io' the angular acceleration of the point. 

The acceleration of B (supposing for the moment A to 
be fixed) will therefore depend on two components, namely, 
the known radial acceleration co^xAB dlongBA and the 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. 157 

known tangential acceleration oj'xAB at right angles to AB. 
Compounding these two accelerations by vector addition, 
we see that the resultant will be AB Vw' + co'^ = a'b\ 
Draw a'l^'=ABXoj^ and parallel to BA, and then /?'6' 
= o)'xAB at right angles to AB and in the proper sense 
given by the sign of the angular acceleration cd\ The 

vector a'6' is thus found; it makes an angle tan- ^—j^ with 

AB. Note that the radial acceleration a'^' is easily found 

(ab)^ 
from the velocity diagram, since its value is ^ . 

The resultant of p^a^ and a'b' is p^b', which represents the 
acceleration of the point B. The line a'6' is in fact the 
acceleration image oi AB, and the acceleration of any point, 
C, on the moving body will be represented by p'c\ where 
the triangle a'6'c' is similar to the triangle ABC. 

Note that in these diagrams the acceleration image 
a'6'c' is the original figure ABC altered in scale, and rotated 



co' 



through an angle j i8o°— tan~ —^ ^ in the sense of co. The 

pole p^ in general does not correspond in position with the 
virtual centre, for p^ represents that point of the body which 
undergoes no acceleration (not that point which has no veloc- 
ity) at the instant considered. 

In the case of the valve-gear of Fig. 97, whose crank is 
rotating with uniform angular velocity, the acceleration 
diagrams shown in Fig. 104 are constructed as follows, sup- 
posing the velocity diagrams to have been previously drawn. 
The acceleration of the point C is wholly radial, since co is 
uniform; thus from the pole p^ the vector p'c^ is drawn 

such that p'c^ = -tt^. Next, the direction and magnitude 

of the radial acceleration of K are known; the line p' z' is 

{peY 
therefore drawn parallel to KB, and of length equal to ~e^d. 



158 



KINEMATICS OF MACHINES. 



The tangential component of the acceleration of E is at 
present only known as to direction; hence e'x' is drawn of 
indefinite length and perpendicular to ^'e'. The point e\ 
which is the acceleration image of E, lies somewhere in this 
line. 



-t>n^8 




50 FT. PER SEC. PER SEC. 



Fig. 104. 



Now consider the acceleration of the point E with regard 
to the link AC, i.e., the acceleration of E round C. Its radial 
component is known; therefore draw cr'e" parallel to EC 

{ceY ' 



(not to CE) and of length 



CE' 



The direction of the tan- 



gential component is known, hence e"^' is drawn at right 
angles to c^€\ and the point e' must lie on this line. Thus 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS, I59 



e' is found at the intersection of e'x' and ^''y\ and p'e' (not 
drawn in the figure) represents the actual acceleration of 
the point E. If cV be produced to d' , so as to be a pro- 
portional copy of CED, d! is the acceleration image of D, and 
p'd' gives the acceleration oiD. 

In the examples drawn out in the figure, pc = 2.^g feet 
per second, while AC = il inch = 0.145 J^^ot. Hence the 

(2.59)^ 
radial acceleration of C is — r^ — - =46.2 feet per second per 

0.145 ^ ^ t^ 

second. 

In the case of the slider-crank mechanism of the direct- 
acting steam-engine, the acceleration diagram is shown in 
Fig. 105. 





Fig. 105. 

The vector p'b^ is first drawn, representing to a con- 

V ^ 
venient scale — ^, the radial acceleration of the crank-pin 

B, and then 6'a' is made equal to the radial acceleration of 
the point A around B ; as found from the velocity diagram 

this quantity is equal to -r^ (see Fig. loi). The line ^V is 

A.Jd 

drawn parallel to AC, and along it will be measured the 
acceleration of A, which has no component at right angles 
to AC. The line «'«' is drawn perpendicular to the direction 
of AB, and a'a' then represents the tangential component 
of the acceleration of A around B. The point a' is thus de- 
termined and the line a'b^ is the acceleration image of the 
connecting-rod AB. 

It will be remembered (§36) that our construction for 
the acceleration of the piston was to set off along BA 



l6o KINEMATICS OF MACHINES. 

a length BF = -^^-r, and to draw FG perpendicular to ABj 

cutting AC in G. It has been pointed out that the line 
GB (see also Fig. 65) is actually an acceleration image of 
AB. This will be plain on comparing the triangles CBG 
and p'h'a^ which are easily shown to be similar, one being 
turned through 180° with reference to the other, so that GB 
and a'6' are parallel. 

53. Example of Polar Velocity and Acceleration Dia- 
grams. — In the case of the Atkinson '* Cycle" gas-engine, 
shown in Fig. 106, we have a good example of the use of 
polar velocity and acceleration diagrams in determining 
the velocity and acceleration of the piston in an engine of 
an unusual type. 

The essential feature of this engine is that for every 
revolution of the crank PA the piston (attached to the point 
D) makes two complete strokes of unequal lengths, its posi- 
tion being shown at points corresponding to the crank 
positions i, 2, 3, 4, 5. This motion is obtained by connect- 
ing the piston to a point Con a link ABC , pairing with the 
crank-pin at A and with a rocking-lever, QB, at B. 

Taking the mechanism in position i, the linear velocity 
of the crank-pin at 180 revolutions per minute being 14.80 
feet per second, we draw the vector pa, representing this 
velocity, and the lengths of ph and ah then give the magni- 
tudes of the velocities of B around Q and of B around A, 
respectively, the directions of these velocities being, of 
course, known. The line ah is the velocity image of AB, 
hence the point c is easily found, remembering that the 
triangles ahc and ABC are similar, but that one is turned 
through 90° with regard to the other. 

We now know pc, the actual velocity of C with regard 
to the frame of the engine. The lines pd, cd are drawn re- 
spectively in the known directions of the velocities of D 
relatively to the frame, and of D about C. They intersect 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS. i6i 




VELOCITIES 


ACCELERATIONS | 


pa 


14.8 


p'a' 


280 


pb 


14.2 


P'^i 


194 


pc 


14.1 


a'32 


0.366 


cd 


10.5 


c'd' 


52.0 


pd 


12.2 


TP'd' 


78 




\\ 




I 
\ 

\ 

\ 








\ ^^ 






\ ^^ 












\ 


\ 


\ 
\ 

\ / \ 

A \ 



l^s 



2 



4 6 8 10 

—I I I I I i_i 



y'—- 

x'-' 



a'\ 



FEET PE8 SECOND 





I I I I 



50 



C' 



150 



FEET PER SECOND PER SECOND 



Fig. io6 



l62 KINEMATICS OF MACHINES. 

Sit d, thus giving the magnitude pd of the velocity of the pis- 
ton along its path. In the example drawn, the velocity of 
the piston is 12.2 feet per second, that of 5 is 14.2, and that 
of C 14. 1 feet per second. The vector cd represents the 
velocity of D with regard to C. 

We have now to draw the acceleration diagram. The 
acceleration of the point A is wholly radial, for the angular 
velocity of the crank is supposed to be uniform. We know 

therefore that this radial acceleration is — '- '-^ =280 

9.4 

feet per second per second, and it acts in a direction parallel 
to AP. Again, we know from the velocity diagram that the 
velocity of B along its path is 14.2 feet per second; B there- 
fore has an acceleration along its radius BQ of — '- 

12.5 

= 194 feet per second per second. Similarly the velocity of 
B about A is found from the diagram to be 0.8 feet per second 
as represented by the length ab, hence the acceleration of B 

along BA is — '- =0.366 feet per second per second, 

and in a similar fashion the radial acceleration of D along DC 

' f ^ ^ u 10.5 X 10.5X12 

is lound to be =52.0 feet per second per 

second. 

Starting from the pole ^', the vector ^'a' is drawn par- 
allel to ^P and of length 280, measured to any convenient 
scale. The real acceleration of the point B with respect 
to A is not known, so that a'6' cannot be drawn directly. 
We can, however, draw a'/?2 =0.366, representing the radial 
component of the acceleration of B with regard to ^. If 
p[b^ is the real acceleration of B, the point b^ must lie some- 
where on a line ^^y' drawn parallel to the direction of motion 
of B with regard to A , and passing through ^2, for a'6' (the 
acceleration of B about A) must be a resultant of the 
radial acceleration a'/?2 and the tangential acceleration whose 
direction is /?2:v'- 



VELOCITY AND ACCELERATION IN PLANE MECHANISMS, 163 

In order to find another line on which 6' must lie, we 
start again from p' and draw p^^^ = ig4, representing the 
radial component of -S's acceleration about Q ; a line /?j^' 
drawn through /?j at right angles to _^'/?j and to BQ gives the 
direction of the tangential component, and it is therefore 
clear that 6' is at the intersection of the lines fi^y^ and fi^x\ 
The vector a'6' then gives the acceleration of B with respect 
to A, and is the acceleration image oi AB. The point c\ 
which is the image of C, is readily found by making the 
triangle a^b^c^ similar to the triangle ABC, and p^c^ gives the 
acceleration of C with regard to the fixed link or frame. 

We may consider this acceleration as being the resultant 
of (i) the acceleration of D with regard to the frame, and 
(2) the acceleration of C with regard to D, We only know 
at present the direction of the first named, and can draw a 
line p^z' parallel to the path of D. The radial acceleration 
of D about C is known to be 52 feet per second per second, 
and the vector c^d^ is drawn to represent this ; note that c^d^ 
must be drawn parallel to DC and not to CD. Through d^ 
a line is drawn at right angles to DC and cutting p^z^ in d\ 
We have then p'd^ for the acceleration of D with respect 
to the frame, and <iV for the acceleration of C about D, 
while d^d^ gives the tangential component and d'c' the 
radial component of this acceleration. Note that c^d^ is the 
acceleration image of CD. The acceleration diagram for 
any other position of the mechanism can be drawn by 
exactly the same method. 

The foregoing examples will serve to indicate the system 
to be adopted in determining the velocity and acceleration 
of any point on a link of a rigid plane mechanism in any 
given position. 



CHAPTER VI, 
ALTERATION OF MECHANISMS. CLOSURE. 

54. Expansion of Elements. — Certain examples have 
already shown the reader how widely the external forms 
of the links in a kinematic chain may be varied, while they 
still retain exactly the same relative motion (see Figs. S;^ 
and 84, in which both mechanisms are the same inversion of 
the same chain). 

We have now to consider further certain cases in which 
links of mechanisms are enlarged, reduced, changed in form, 
added, or omitted, without altering the relative movements 
of other links. 

Perhaps the most familiar instance of a change in form, 
which in this case is really the expansion of an element, is to 
be found in the eccentric so generally employed for obtain- 
ing a reciprocating from a rotary movement in valve -gears 
and elsewhere, and shown in Fig. 97. 

Let us suppose in a slider-crank chain that while the cen- 
tres of the links remain the same, the radius of the cylindrical 
surface of the turning pair ah is increased, as in Fig. 107, 
until at length the crank becomes a disc, inside of which lies 
the centre of the pair ad. 

The crank a has now taken the form of an eccentric, 
without in any way changing the relative motion of the links, 
the only alteration being that one element formed on each 
of the links a and h has been expanded . 

Again, take the case already mentioned (in § 34) where 

in the quadric crank -chain a, swinging link c has appar- 

164 



ALTERATION OF MECHANISMS. CLOSURE, 



165 



ently been replaced by a sliding block travelling in a curved 
slot, as shown in Fig. 6oh. Notice that the pair he remains 
just as before, while the appearance of the chain (but not the 
relative motion of its links) has been changed simply by 




Fig. 107. 



increasing the radius of the turning pair cd, and utilizing only 
a portion of its curved surface. The effective or kinematic 
length of the link c remains unaltered. 

As a third example, imagine the radius of the pair be. 



i66 



KINEMATICS OF MACHINES. 



in the slider-crank chain of Fig. io8, to be increased as 
shown, until it is greater than the length of the link h, while 
the link a retains its original form, that of a crank. The 
expansion may be carried a stage farther, as shown in Fig. 
109, by increasing the radius of the pair ab also, but in a 




Fig. 108. 




Fig. 109. 

lesser degree. The links a and h have now both become 
eccentrics, while c takes the form of a strap provided with 
projections sliding in guides formed on d. The kinematic 
lengths, however, of the links in Fig. 108 are just the same 
as in Fig. 109, and the relative movements are the same. 



ALTERATION OF MECHANISMS. CLOSURE. 167 

Here we have instances of the expansion of the pairs of 
elements be and ah. 

55. Augmentation of Chains. — Many instances occur in 
which typical kinematic chains are apparently disguised by 
the introduction of additional links. Such a change is 
called by Reuleaux the augmentation of a chain or of a 
mechanism, and the links which are added, while giving 
the chain no new kinematic properties as a whole, are intro- 
duced for convStructive reasons. 

Take as an instance a bicycle wheel and its axle. Here 
the movement of the wheel relatively to the frame to which 
the axle is attached is exactly the same as if the connection 
between them were a simple turning pair. But on examina- 
tion we find that the actual pairing is of quite a different 
character, and that a series of balls running in grooves cut 
in the hub of the wheel, and in the axle, have been provided, 
to minimize friction and wear.* 

Some examples occur in which a relative motion that 
might have been attained by simple pairing is arrived at by 
the use of a whole chain, and such cases might equally well 
be looked upon as instances of augmentation. 

A complex train of toothed wheels is often employed to 
give a velocity ratio which might have been obtained by a 
chain much simpler mechanically, but occupying more 
space, or inadmissible for some other reason. Again, in a 
steam-engine indicator the piston and rod are guided, not 
by a simple sliding pair, but by a straight-line motion, which 
is in itself a kinematic chain. Many other instances might 
be cited in which pairing is replaced by '* chaining," that is, 
by the introduction of linkage ; the link or links introduced 
being so arranged as not to alter any of the relative motions 
already existing. 

It has been pointed out by Reuleaiix that a chain which 
alread]| possesses the largest number of links which it can 

* See Fig. 225. 



i68 



KINEMATICS OF MACHINES. 



have as a simple closed chain, necessarily becomes a com- 
pound chain on augmentation. 

We shall see later that chains have occasionally to be 
thus augmented for purposes of closure. 

56. Reduction of Chains. — A reduced mechanism is ob- 
tained by the omission of one or more of its links, corre- 
sponding alterations being made in the pairing of the remain- 
der. To illustrate this we may take the gear used for 
actuating the valves of duplex non-rotative steam-pumps, 
which has been already mentioned and is shown in Fig. no. 

This mechanism is essentially a reduced crossed-slide 
chain (a crossed-slide chain is shown in Fig. 86, § 45). 

Using the same letters as in that figure, we see that the 
link h has been omitted, the lever a has its lower end formed 
into a figure of constant breadth measured in the direction 
of motion of c, and that end is fitted easily between two 
parallel lugs or projections on c. It is plain that the relative 
movement of c and a is practically unchanged.* 




Fig. 1 10. 

We have here, as a result of omitting the link h, the intro- 
duction of higher pairing with its advantages and disadvan- 

* This statement is correct only for small displacements of the link xi from its 
mid-position. 



ALTERATION OF MECHANISMS. CLOSURE. 169 

tages, among the latter being the mechanical trouble which 
will arise when the material of the links wears and the fit 
of the lever end becomes slack. The reader will find it easy 
to discover for himself numberless similar examples of aug- 
mentation and reduction in machinery which he has the 
opportunity of examining. 

57. Reduction by Use of Centrodes. — Interesting cases 
of reduction occur sometimes in which chains are reduced 
by the introduction of higher pairing between new links 
taking the form of centrodes of links of the original mech- 
anism. This can only be completely carried out where the 
centrodes take the form of closed curves. Consider, for 
instance, the form of the quadric-crank chain in which oppo- 
site links are equal, while the longer links cross one another. 
This chain and its virtual centres are shown in Fig. ma. 
It is evident from symmetry that the sum of the distances 
from ^^ and O^^to 0^^ must be constant, and equal to the 
length of one of the longer links. Hence, if a or c is the 
fixed link, the path of 0^^ will be an ellipse of which the 
length of the link a or c is the focal distance. The two 
ellipses will always touch at the point (0^^) which describes 
them, and, in accordance with the well-known property of 
centrodes, may be imagined to roll on one another as the 
links move. Now suppose that d is the fixed link, and 
imagine that elliptical plates in the form of the pair of 
centrodes are attached to a and c (Fig. 11 16). On removing 
the link h, we then get a mechanism (Fig. iiic) of three 
links only, and if proper constraint were applied * the 
links a and c would roll on one another, at the same time 
having exactly the same angular velocity ratio as if directly 
connected by a link h. We have thus reduced the mechan- 
ism and introduced higher pairing without affecting the 
relative motions of the remaining links. 

* See §§ 3 and 4. 



lyo 



KINEMATICS OF MACHINES. 



The employment of links shaped thus in the form of 
centrodes must not be confused with the method of trans- 
formed or reduced centrodes adopted by Reuleaux for the 



CENTRODE OF tt ( C FIXED ^ 




Fig. II iA 



Fig. II I r. 



purpose of expressing the velocity ratio of certain links in 
the simpler mechanisms.* 

It is worthy of notice that the second pair of centrodes 

* Reuleaux, Kinematics, p. 70. 



ALTERATION OF MECHANISMS. CLOSURE. 171 

in the anti-parallel crank mechanism (those of the two longer 
links) are a pair of hyperbolas. The reader should draw 
these as an exercise. 

58. Closure of Incomplete Pairs. — The meaning of the 
term closure as applied to pairs and to kinematic chains was 
explained in §§ 3 and 4. We have now to discuss the 
methods of applying such closure or restraint in various 
cases in which the pairs or chains would otherwise be incom- 
plete. 

On examination it is found that in a large number of 
pairs of elements existing in actual machines the forms of 
the elements are not such as to completely constrain their 
relative motion. For example, in certain forms of axle- 
boxes for cars or locomotives we find that the brass of the 
bearing embraces only a comparatively small angle on the 
upper surface of the journal, and hence the form of the 
bearing does not render separation of the surfaces in con- 
tact im.possible. Such separation does not occur in practice, 
for the reason that the weight of the car presses the brass 
against the journal. We have here an example of force- 
closure of a pair. It is often necessary and convenient to 
employ force-closed pairs, since they are so readily taken 
apart for examination, and are usually simple in form. 
The table or platform of a weighing machine, for instance, 
generally rests on its knife-edges without being held down 
in any other way than by gravity. The force of gravity is 
not the only one employed for closure of pairs ; in friction- 
gearing, for example, the rollers must be pressed together 
by some external force, so that one wheel can drive the 
other without slipping. We shall find also that force- 
closure has very frequently to be applied in pairs involving 
non-rigid links. 

Although, strictly speaking, all pairs in mechanisms 
must be closed, either by the forms of their surfaces or 
by the application of an external force, cases occur in which 



172 



KINEMATICS OF MACHINES. 



the desired object is attained by making the pair of elements 
into a kinematic chain complete in itself, in which the added 
links have for their only object the provision of the necessary 
restraint. This chain closure is mxore generally applied for 
the purpose of constraining the motion of incomplete chains, 
as will be presently seen. 

59. Closure of Incomplete Chains. — As in the case of a 
pair, we may say that a kinematic chain is incomplete if the 
relative movements of its parts are not completely defined. 




Ftg. 112. 

A chain may be incomplete (a) because it contains too 
many links, or (6) because it has not enough pairs of ele- 
ments, or (c) either' dead-points or change -points (§ 30) 
occur at which it is locked, or its motion becomes indeter- 
minate. 

Closure may be applied in an incomplete chain (i) by a 
force, (2) by the duplication of the mechanism, (3) by the 
addition of a pair of elements or another link. 

When the incompleteness of a chain consists solely in 
"the incompleteness of a certain pair, this may be rectified 



ALTERATION OF MECHANISMS. CLOSURE. 173 

by any of the methods just discussed. We need only con- 
sider therefore how to treat chains consisting entirely of 
closed pairs, the motion of which is indeterminate on accoimt 
of the existence of dead-points or of change-points. 

The flywheel of a single-cylinder steam-engine is an 
example of the use of the first method, force-closure, in 
passing dead-points, for it is the energy already stored up 
in the flywheel which keeps the crank rotating in positions 
where the steam pressure is not able to exert any turning 
moment on the shaft. 

The energy of a moving body, such as a flywheel, is 
evidently not available when the machine to which it is 
attached is only moving very slowly or is just about to 
start. In these cases the second method, chain-closure, 
must be employed. For example, in a locomotive, or in a 
marine steam-engine, the mechanism is so arranged that 




Fig. 113. 

when one crank is at its dead-point, another, actuated by 
a separate steam-cylinder, piston, and connecting-rod, is 
in a more advantageous position. The original chain has 
thus been closed by duplication of the mechanism. 

Another instance may be taken. In the well-known 
three-cylmder or Brotherhood type of single-acting steam- 
or hydraulic -engine (Fig. 112), three complete sets of driv- 
ing apparatus work on the same crank, with the result that 
only one set can at any mstant be passing its dead-point. 



174 



KINEMATICS OF MACHINES. 



The parallel-crank mechanism of Fig. 113 is an instance 
of closure by the employment of a fifth link, which is adopted 
to enable the chain to move past a position which is not only 
a dead-point, but also a change-point. It has been already 
mentioned in § 30 that the necessary closure is obtained 
in this mechanism, when used on locomotives, by the addi- 




b. 









iiJ 


K 




111 


_j 


^ ^ 


III 

a. 

\- 
z 


-1 
> 



u 














Ul 


e 


z 


Q 




z. 


Ul 


J 




> 




u 


z 




Ul 














Fig. 114. 

tion of another pair of cranks in another plane and a second 
coupling-rod. Fig. 113 represents a model which illustrates 
this arrangement. The same object may be attained by 
the placing of a third crank in the same plane as the first 
two, when the coupling rod takes the form of a triangular 
plate; thus a fifth link has been added. Fig. 114 shows 



ALTERATION OF MECHANISMS. CLOSURE. 175 

the link- work of a form of steam-engine * in which dead- 
points are avoided by the addition of new hnks and the 
partial duplication of the chain. Here the connecting-rod 
h is transformed into a triangular frame, and is paired 
(i) with the crank a, (2) with a link, e^, connecting with 
the crosshead c^, (3) with a link, e^, attached to a second 
crosshead c^, and (4) with a guiding link, /, attached to a 
fixed point on the frame. It will be seen that when the pis- 
ton which actuates c^ is unable to exert any turning moment 
on the shaft, the piston of c^ is very nearly in the most favor- 
able position with regard to the crank. In this way an 
engine is obtained which has no dead-point, and which has 
a very uniform turning moment compared to that of the 
ordinary single-crank engine. The fixed link or framework 
of the engine, d, is not shown in the diagram. 

On consideration, it is obvious that in a chain which has 
change-points, but is otherwise closed, we can obtain com- 
plete closure if pairs of elements are arranged corresponding 
in form to the required motion, and coming into action or 
constraining the motion of the chain at the required in- 
stants. Such pair-closure of chains is, of course, the con- 
verse of the chain-closure of pairs, which has been already 
discussed. We shall see in a later chapter how to form a 
pair of elements, in general, so that they may have any 
desired relative plane motion, and shall find that most fre- 
quently such elements will have higher pairing. 

As an example of pair-closure of a chain, one method of 
closing the anti-parallel crank chain at its change-point is 
shown in Fig. 115. 

A pin P and a gab G are placed on the links h and d 
respectively, so that they are in contact at the proper in- 
stant, and permit only one kind of motion at the change- 
point. To do this the pin P must be placed on h at the 
point where it is cut by the centrode of d with regard to 6, 

* See Engineering, October 28, 1892. 



176 



KINEMATICS OF MACHINES, 



and the gab must be where d is cut by the centrode of h 
^vith regard to d. 

The pin and gab may be considered as portions of the 
two centrodes, appHed for the purpose of closure as in Fig. 
Ill, and so shaped as to prevent any relative slipping. 
In a similar manner the links a and h might be paired by 
using as pins and gabs portions of their elliptical cen- 




FiG 115. 



trodes. Such centrodes would be equivalent to small por- 
tions of elliptical wheels, toothed to prevent slipping. 
Many instances occur in which pairs of elements are 
thus applied so as to act at the change -points of a mechan- 
ism, and prevent its transformation into another mechanism 
or into a pair of elements. In general, pair-closure must be 
provided for each change-point ; in the mechanism of Fig. 115 
the two pins and two gabs are required so as to come into 
action at the two change-points occurring in each complete 
revolution of the chain. 



CHAPTER VII. 

CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING 
INVOLVING PLANE MOTION. 

60. Constraint of Bodies having Plane Motion. — It has 

already been stated that a body free to move in a plane 
possesses three degrees of freedom and has three degrees of 
constraint. Further constraint may be applied by causing 
such a body to touch certain points on the surface of a sec- 
ond rigid fixed body, these points being known as points 
of restraint. A point of restraint of a figure or body may 
be defined as a point on its outline, so touched by a point 
on the outline of a second fixed figure or body, that no rela- 
tive sliding motion is possible along or parallel to the com- 
mon normal to the two figures at the point of contact. 
When thus restrained the body or figure is considered as 
being kept in contact with the pomt or points of restraint. 

We may take an example to illustrate the meaning of 
this definition, and to show the actual nature of points of 
restraint. Suppose (in Fig. it 6a) that it is required to ar- 
range a support or base, a, for a tripod, h, so that an instru- 
ment fixed on h can be removed from its support and re- 
placed exactly in its previous position. This may be effected 
by providing h with three rounded points or legs, CDE. A 
hole, F, is made in the base, a, and is of pyramidal or conical 
form, so that if the rounded end of C is placed in F, there 
will be contact at three points of restraint ; in this way, so 
long as the contact is maintained, the only possible relative 
motion of c and a will be one of rotation about some axis 

177 



178 



KINEMATICS OF MACHINES. 



passing through the centre of the spherical surface of the 
end of C. The next step is to provide on a a slot or groove, 
G, of triangular cross-section as shown ; when D is placed in 
this groove there will be tw^o more points of restraint, and 
the only possible relative motion remaining will be a rota- 
tion about the axis CD. Finally the position of h is fixed 
relatively to a if the third point E is made to rest in contact 
with a flat surface, H, formed on or connected with a, thus, 
furnishing the sixth point of restraint required (see § 7).. 




Fig. ii6a. 



The whole device is known as the * ' hole, slot, and plane.' * 

The application of similar principles is illustrated in the 
design of Ewing's extensometer,* an instrument for meas- 
uring the deformation of test-pieces under stress. In this 
apparatus the bar or test-piece whose extension or com- 
pression is to be measured (a in Fig. 1166) carries a clip, 6, 
attached by the points of two set-screws in such a way that 
h can move relatively to a about the axis of the set-screws 
at B. The clip h carries a projection, U , ending in a rounded 

*Ewing, Strength of Materials, p. 75. 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. I79 

point F. This point engages with a pyramidal or conical 
hole formed on a second clip, c, which is also secured to a by 
means of two set-screws at C. So long as F rests in its 
recess, h and c can have no relative motion unless the length 
BC alters ; in that case the angular motion of h and c will be 
proportional to the extension or compression of a. Actually 
the projection h' is not rigidly attached to 6, but can turn 




Fig. iidb. 



through a small angle about the axis BD. This provision 
is made in order that any rainute twist of the test-piece a 
about its axis BC may not affect the angular motion of h 
and c to any appreciable extent. This angular movement 
is indicated by the scale E attached to c ; the distances CF, 
CG are equal, so that the movement of the scale, as read by 
the microscope at M, will be twice the actual deformation 
of the test-piece as taken on the length BC. 



i8o 



KINEMATICS OF MACHINES. 



Similar methods are followed in designing the so-called 
kinematic clamps and kinematic slides.* 

A kinematic clamp is a contrivance intended to fix com- 
pletely the position of one body with reference to another ; 
a kinematic slide permits one body to have on." degree of 
freedom with reference to another. 

On consideration it is plain that in a kinematic clamp 
or slide the points of restraint must be suitably placed 
with regard to the shape of the body to be restrained. 

It is thus proper to inquire what must be the disposition 
of the points of restraint required, either to define the posi- 
tion of one body relatively to another, or to permit the 
movable body to retain one degree of freedom, and thus to 
constrain its motion completely. We shall suppose that 
the movable body at first possesses three degrees of free- 
dom, and is capable of plane motion. 

Let a (Fig. 117) be such a body, and let a fourth point 





Fig. 117. 



Fig. 118. 



of restraint, A, be provided, in addition to the three points 
necessary for insuring plane motion. The arrow-head 
then represents the fourth point of contact of the restraining 
or fixed body. 

Draw A A' normal to the tangent of tlje outline of a at A. 

Any possible motion of a may be regarded as an instan- 



*For an example of a kinematic slide, see Min. Proc. Inst C. E., Vol. 
CXXXll, F 49- ' 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. i8i 

taneous rotation about a virtual axis perpendicular to the 
plane of motion. We need therefore only consider how the 
single point of restraint, A, affects the possibility of turning 
the body a about such an axis. It must be remembered 
that by the definition of a point of restraint, a is to be kept 
in contact with the restraining body. This is impossible if 
the virtual centre is not somewhere along AA' , for if the 
virtual centre were, say, at B, a, point about which only 
right-handed rotation is possible, it is plain that such rota- 
tion could only occur if the point A ceased to touch the 
restraining body. Hence we see that any possible instan- 
taneous motion of a must be about a virtual centre situated 
in A A', and any motion of translation must be along a line 
at right angles to AA\ 

Next consider the effect of keeping the body a in contact 
with a restraining body at two points, A and B. Let the 
normals AA^, BB' intersect at P. The body is then only 
capable of an instantaneous rotation about P. If the nor- 
mals are parallel, then only an instantaneous motion of 
translation, i.e., rotation about an infinitely distant axis, 
will be possible. 

On adding another point of restraint, C (Fig. 119), it will 
be found that if we suppose that the body a remains in con- 
tact with the three new points of restraint, A, B, C, no 
movement is possible, except when the three normals inter- 
sect in one point or are parallel. In these cases instanta- 
neous turning abqut the point of intersection, and instanta- 
neous translation about an infinitely distant axis, are re- 
spectively possible, so that a at the instant considered will 
thus possess one degree of freedom and will have constrained 
motion. 

In Fig. 119 a little consideration shows that no move- 
ment at all is possible except about an axis situated within 
the triangle PQR, so long as the restraining body is rigid. 
The whole field of motion, with the exception of PQR, then 



l82 



KINEMATICS OF MACHINES. 



becomes what Reuleatix calls a "field of restraint." But 
if movement did occur about an axis placed within the 
triangle PQR (in the figure such rotation could only be 
right-handed), the body a would at once cease to touch the 
restraining points with which we suppose it to be kept in 
contact. A similar result will be found with other arrange- 
ments of the points of restraint, and therefore in general 





Fig. T19. 



Fig. 120. 



the position of a will be fixed if it is made to touch the re- 
straining body at the three additional points A, B, C, a 
result already stated in § 7 . 

Fig. 120 shows the case in which the three normals meet 
in a point, P. If the shape of the body a is such that no point 
of restraint can be so apphed as to have a normal that does 
not pass through P, then the body cannot be fixed by the 
application of three or any number of points of restraint, 
and its shape must be altered for that purpose. For exam- 
ple, a circular disc having plane motion could not be so fixed. 

It is thus evident (a) that if one of two rigid bodies 
capable of relative plane motion remains in continuous con- 
tact with two points of restraint formed on the second body, 
the relative motion is constrained, and the virtual centre of 
the two bodies is always at the intersection of the two com- 
mon normals. 



CONSTRAINT AND yELOCITY RATIO IN HIGHER PAIRING. 183 

Also (6) if three points of restraint are employed, and 
contact at all three is continuous, constrained relative 
motion is only possible if the three common normals inter- 
sect in a point, or are parallel. 

The reader will find that the constraint of the motion 
of a body by means of such points of restraint as have been 
defined above is an easier matter than the limitation of the 
movement of a body by points of contact with a second 
fixed body, if no force is supposed to keep the two bodies 
in contact. In this case the bodies would possess greater 
freedom of motion than under the restrictions we have 
supposed. The theory of constraint has been treated by 
Reuleaux * and by Burmester,t to whose woiks the student 
is referred for information on the subject. 

61. Closed Higher Pairs having Plane Motion. — Let us 
next suppose that the moving body a and the second or fixed 
body 6, while kept in continuous contact, have such forms 
that one is the geometrical envelope of the other, and that 
in every position the normals at the several points of con- 
tact are either parallel or meet m a point. It is obvious that 
in this case at any instant a can move in one way, and in 
one way only, with reference to h; in other words, a and 6 
will form a closed pair AVe proceed to consider some exam- 
ples of such pairing, w^hich in general will be higher pairing, 
in accordance with the definitions m § 2. 

In Fig. i2ia, let ABCD be a figure (called by Reuleaux 
a Duangle), drawn by describing the arcs ABC, CD A, with 
a radius equal to BD, and with D and B as centres re- 
spectively. Suppose that this figure, representing a body, a, 
having plane motion, is made to touch two lines, PQ, QR, 
inclined at an angle of 60°, the points E and F on these 
lines forming points of restraint for the duangle, and the 
lines PQ, QR representing the profile of the restraining 
body b. The normals at E and F to QR and QP will inter- 



* Reuleaux, Kinematics. Chapter III. f Burmester, Kinematik, Chapter V. 



i84 



KINEMATICS OF MACHINES. 



sect at 0, where they make an angle FOE = 120°, and they 
must pass respectively through the points B and D, since 
these points are the centres of the arcs ADC, ABC, 

As the duangle moves in contact with PQ and QRy the 




CENTRODE OF 6 
(a FIXED) 






Fig. i2i3. 



path of ^ must be a straight line, GBH, parallel to QR and 
at a distance, BE, from it. The path of D similarly must 
be a line, GDK, parallel to QP, Hence the motion of the 



CONSTRAINT AND yELOCITY RATIO IN HIGHER PAIRING. 185 

duangle relatively to PQ, QR will be the same as that of a 
straight line of constant length, BD, whose ends lie contin- 
ually upon two lines, KG, HG, enclosing an angle of 60°; 
further, the virtual centre of the two bodies will be the 
point 0, the intersection of the two common normals. 

Since the angles OBG, ODG are right angles, a circle may 
be drawn on GO as diameter, passing through the points 
GB, OD. The point A also lies on this circle, since the angle 
BAD is 60°. Join AG. Then so long as the curves ABG, 
ADC touch the lines QP, QR respectively, the angle AGD = 
angle ^5D = constant. Thus A lies continually on a line, 
RP, drawn through G and inclined at 60° to RQ. PQR is 
then an equilateral triangle, inside of which the duangle 
moves. The relative motion of the triangle and the duangle 
will be constrained if A is the normal to PR at A; i.e., if 
the three normals at the points of contact meet at 0. This 
is seen to be the case, for the angles AOD, ABD, ADB, 
AOB are all equal. Hence AO bisects the angle BOD and 
is perpendicular to PR. 

The path described by with reference to the triangle 
PQR is the centrode of the duangle. It evidently consists 
of a curve joining K and H. Now in any position the circle 
drawn on GO as diameter and passing through B and D has 
a chord, BD, of constant length, and the angle BGD is con- 
stant. Hence GO, the diameter of this circle, is the same 
{length = GH) for all positions of 0. Thus lies on a cir- 
cular arc joining K and H and having G as centre, and the 
complete locus of with regard to the triangle is an equi- 
lateral curve-triangle GKH (Fig 121b). Since the angle BOD 
is constant, the locus of with regard to the duangle is seen 
to be a duangle BODO\ the radius of whose sides is iGO. 

The whole relative motion of the duangle A BCD and 
the triangle PQR is thus represented by the rolling of the 
duangle BODO\ inside the curve-triangle GHOK. The 
centrode of A BCD with regard to the triangle PQR is GHOK ; 



1 86 KINEMATICS OF MACHINES. 

that of the triangle with reference to A BCD being BODO'. 
Any point on the duangle A BCD will have a path made 
tip of trochoidal curves described on the plane of the 
triangle PQR, and vice versa. 

Relative motion of the duangle and the equilateral tri- 
angle may evidently be represented by the rolling together 
of a pair of circular arcs, one having a radius twice that of 
the other. Points on either figure will therefore describe 
trochoidal curves on the other. 

The example just given will indicate the method of 
studying the relative motions of the elements of higher 
pairs having plane motion. A large number of closed 
higher pairs may be devised by utilizing figures of constant 
breadth. The equilateral curve-triangle previously men- 
tioned is such a figure, and its motion relatively to a circum- 
scribed square may be followed as an exercise. 

A number of other forms are given by Reuleaux in the 
chapter already quoted. The student should note in all 
these cases that the form of the path described on 5 by a 
point on a is not the same as that described on a by the 
corresponding point on 6, a condition previously mentioned 
as being characteristic of higher pairing. 

62. Form of Elements for a Given Motion. — Having illus- 
trated the method of determining the centrodes and the 
relative motion in the case of higher pairs of mutually re- 
straining elements of given profile, we have next to show 
how to solve the converse of this problem, namely, how to 
find the forms of a pair of elements whose relative motion is 
previously decided. The relative motion in question must, 
of course, be defined by the forms of a pair of given cen- 
trodes, the mutual rolling of which, as already stated, rep- 
resents the relative motion required. It most frequently 
happens in practice that we have also given the form or 
profile of one element of the pair, and the form of the second 
has to be found. 



CONSTRAINT AND yELOCITY RATIO IN HIGHER PAIRING.i^l 

Let A A and BB (Fig. 122) be a pair of centrodes, of 
which A belongs to, or is traced upon, a body whose profile 
is aa\ It is required to find the profile of a second body to 




which the centrode BB belongs ; the profile to be such that 
while the two bodies remain in continuous contact the cen- 
trodes will roll on one another and the bodies will thus have 
the desired relative motion. 

Take any point, a, in the profile aa^ and draw aC normal 
to aa^ at a, and cutting the centrode A at C. In this case 
for convenience aa^ is shown in the figure as a straight line, 
but it may, of course, be of any form. 

At the instant when the profile 66' (to be found) touches 
the given profile at the point a, aC must be the common nor- 
mal, and the virtual centre of the two bodies must lie on this 
normal, for otherwise contact would not be continuous. 
The point C, where the normal at a cuts the centrode A, 
must at that instant be the virtual centre of bb' with regard 
to aa^f since the curve AC is the locus of the virtual centre 
of 6. AC may be regarded as being attached to aa^, since 
it is a curve traced on the body represented in outline by aa\ 
We proceed to find a point, 6, on the profile of the second or 



1 88 KINEMATICS OF MACHINES. 

moving body, such that when a and h are in contact C is 
the virtual centre of the two bodies and aC the common 
normal. 

Suppose that the centrodes A A and BB are in contact 
at some point D, and measure along the centrode BB a 
length DC^ equal to the length of DC measured along AA. 
Draw B^CD^, representing the centrode B in the position it 
occupies when a is the point of contact of the two bodies and 
C is their virtual centre, and make CD^ = CD. Join aD^. 

Then since the outline of bb^ may be regarded as attached 
to the centrode B, any point on that outline having the same 
position in relation to C^ and D that the point a has in rela- 
tion to C and D^ will be the point that touches a when the 
centrodes touch at C. Accordingly we need only make bD = 
aDj and bC^=aC in order to determine the position of b. 
The point b is then a point on the required profile which 
will touch the point a when C is the virtual centre of the 
two bodies. In the same way we can determine any other 
point on the profile required, and it only remains to pro- 
vide the resulting body with the restraint required to pre- 
vent any other motion than that desired. This would in 
general be done by so forming the body bU that it possesses 
at any instant three points of contact with aa\ the normals 
to these points always intersecting at the virtual centre. 
It would, in fact, be necessary to repeat the construction of 
Fig. 122, assuming two other portions of the outline of aa\ 
and finding two new portions of the outline of bU , the cen- 
trodes, of course, remaining the same as before. It may 
be noted that while the relative motion of the centrodes 
is one of simple rolling, that of the two outlines is in general 
rolling and sliding combined. 

63. Condition for Uniform Velocity Ratio. — We have seen 
in § 60 that when two bodies are in continuous contact and 
are capable of constrained relative motion, the normals at 
the points of contact must intersect at the virtual centre. 

Consider now the case of three bodies (Fig. 123) of which 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 189 



acf 



0^^, with refer- 



a and b turn around permanent centres 

ence to the fixed body c, the bodies a and b being connected 

by higher pairing at a point of contact, P. The three bodies 




Fig 123. 

thus form a kinematic chain of three Hnks, and we know 
that their virtual centres must He in a straight hne. Hence 
0^^ hes on the Hne joining 0^^ and 0^,^. But the normal at 
P must pass through 0^^, if contact is to be continuous. 
The position of 0^^, is thus fixed, and if a and b are to have 
one degree of freedom, the normal at any other point of 
contact, Q, must likewise pass through 0^^. 

Next suppose that at the instant considered a has a cer- 
tain angular velocity, io^^, with reference to c, in consequence 
of which any point on a will have an instantaneous linear 
velocity measured by the product o;^^X radius. The point 
^ab is ^ point common to a and b, and its linear velocity 
will be 0^ ac^^ aP ab^ ^^^ dircctiou being of course perpendic- 
ular to the radius, i.e., perpendicular to the line O^fi^^. 
Knowing the linear velocity of 0^^ we can find the angular 
velocity of b, which must of course be 



^6c = 



linear velocity oj^^ X O^fi^^ 



radius 



o.,o,. 



k 



190 



KINEMATICS OF MACHINES. 



Hence 



angular velocity of a o)^ 
angular velocity of 6 oj 



o.fi,. 



be 



Oafi,, 



In other words, the virtual centre ^^ divides the distance 
^afibc inversely in the proportion of the angular velocities 
of h and a with regard to c. From this important result it 
follows that if the angular velocity ratio for the bodies a and 
h is to be constant, 0^^ must be a fixed point, in which case 
its path on each of the bodies a and h must be a circle. 
Uniform angular velocity ratio will then involve the rolling 
together oi circular centrodes. 

If we desire to find the angular velocity of a with regard 
to h (that of h with regard to c being known) we have only 
to suppose h fixed, and it follows that 



OJ 



ah 







b<^ 



OJ 



cb 



OaPab 



64. Wheel-gearing. — It may be noticed that if we form 
a and h in the shape of their centrodes we get a means of 
connecting two shafts (fixed to these bodies) with uniform 
angular velocity ratio, and the angular velocities, as we have 
just seen, will be in the ratio of the radii of the circular cen- 
trodes r^ and r^, supposing that these centrodes roll together 
without slipping. 

Thus in Fig. 1 24 we should have 

Q^J_b 
PR r • 



OJ. 



OJ 



be 




Fig. 124. 
Such kinematic chains, used for the purpose of trans- 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 191 

mitting motion from one shaft to another with any desired 
uniform or variable velocity ratio, are known as wheel - 
trains or gear-trains ; the shafts, as we shall see in a later 
chapter, need not be parallel, but may intersect or even may 
not meet at all, the gearing being arranged to suit these con- 
ditions. 

It is sometimes possible to use the actual forms of the 
centrodes (or, more strictly speaking, axodes) as the shapes 
of the gear-wheels. For example, the centrodes of the short 
links of an anti-parallel crank-chain might be replaced by 
elliptical gear-wheels, as we have already seen in § 57, and 
if these wheels rolled together without slipping, the shafts 
attached to them would at every instant have the same 
velocity ratio as the short links of the original chain. In 
the same way we can design many other forms of gear-wheels 
which when rolled together without slipping will transmit 
motion from one turning pair to another with some desired 
uniform or variable velocity ratio. Such wheels in the form 
of smooth axodes are not very useful in practice on account 
of their liability to slip; it is therefore usually necessary 
to provide the surface of each with teeth. These teeth 
have higher pairing, and their relative motion is in general 
combined rolling and sliding. The form of their profiles can 
therefore be determined by the method already explained 
in § 62, the two centrodes and the form of one profile being 
given. We shall return later to the question of the forms of 
wheel-teeth. 

Wheels having such forms that their outlines are their 
own centrodes (or, more correctly, their surfaces are their 
own axodes), take a great variety of forms. It will be suffi- 
cient to notice here only cases in which the axes of rotation 
are parallel, and the planes of motion of the wheels therefore 
coincide. The elliptical wheels already mentioned afford 
one example, but other forms,, usually termed lohed wheels, 
are occasionally met with. In every instance the point of 



192 



KINEMATICS OF MACHINES. 



contact of the centrodes, i.e., the virtual centre of one wheel 
with regard to the other, must lie somewhere on the line of 
wheel centres. In a pair of lobed wheels, the difference 
between the greatest and least radii of the centrodes is called 
the inequality; in elliptical wheels the inequality is equal 
to the focal distance of either eUipse. Properly shaped 
lobed wheels working together have the same inequality. 
Thus in Fig. 125 the difference of PA and PB, the greatest 




Fig. 125, 

and least radii of the three -lobed wheel a, must be the same 
as QD — QC, the inequality of the two-lobed wheel b with 
which a gears. Further, the outlines of the wheels must be 
such that in every position of contact the point R lies .some- 
where on the line PQ, for if this were not so R could not 
be the virtual centre of a with regard to 6. 

In practice lobed wheels find only a very limited appli- 
cation, and the reader is referred to other works for infor- 
mation as to the shape to be adopted in any particular case.* 

* See Rankine's Machinery and Millwork, p, 97 ; MacCord, Kinematics of 
Mechanical Movements, ^98. 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 



193 



65. Spur-wheels. — Wheel-gearing is most frequently 
employed to connect two shafts whose axes are parallel and 
whose angular velocities are in a constant ratio. It is some- 
times sufficient to use friction gearing (in the form of smooth 
or grooved circular rollers), but gearing of the kind now 
to be discussed is in general provided with teeth lying 
parallel to the axes of the wheels, and is known as spur- 
gearing. The circular centrodes are called the pitch circles 
of the wheels, for it is around their circumferences that the 
teeth are set off. 

Let a and b (Fig. 126) be portions of two spur-wheels 
gearing together with uniform velocity ratio, c is the fixed 




Fig. 126. 



link of the chain, and P and Q are 0^^^ and 0^^ respectively. 
The point R{ = 0^^) is the point of contact of the centrodes 
or pitch-circles and is called the pitch-point. 

Let 5 be the point at which the profile of the tooth formed 
on a is in contact with the profile of the tooth formed on h. 
Let SR be the common normal to the tooth profiles at 5, the 
point of contact. At the instant considered, V, the velocity 
of 5 resolved along SR, must be the same whether we con- 



194 KINEMATICS OF MACHINES. 

sider 5 as a point on a or as a point on h. The actual 
motion of 5 (as a point on a) is that due to a velocity V ^ 
in a direction at right angles to PS. Similarly the actual 
velocity of 5 (as a point in h) must be V^, at right angles 
to QS. It is plain that if contact is to be maintained at 5 
during the instant considered, V ^ and V^ must have the 
same component V along the common normal to the two 
surfaces at 5. We may in fact regard 1^^ as the resultant 
of two velocities, V along the normal and v^ perpendicular 
to it ; similarly V^ is the resultant of V and v^. 

Evidently v^^ — v^ measures the speed at which the sur- 
face of b is sliding relatively to that, of a ; one object in a 
well-designed gear should be to make this sliding motion 
as small as possible, so as to minimize wear. 

V 
The angular velocity of a is measured by the ratio -p° 

V 
which from the figure is seen to be equal to pjir, where 

PM is the length of the perpendicular dropped from P on 

V 
RS. In the same way yrj- measures the angular velocity 

of b, and 

<^j. V _ V _QN __ QR 

'be 



a ' 



ac 

oj, PM ' QN PM PR' 



which is constant if i? is a fixed point in the line PQ. 

We thus see that for uniform velocity ratio the forms of 
teeth must be such that the common normal at the point 
of contact always passes through a fixed pitch-point, R, 
which divides the line of centres in the inverse ratio of the 
angular velocities, and is in fact 0^^. This important result 
has already been obtained in a more general manner in 

§63. 

Two kinds of curves at once suggest themselves as being 
suitable for wheel-teeth profiles, because their normals are 
easily found. These are involutes of circles and the vari- 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 195 

Oils cycloidal curves produced b}^ rolling one circle, attached 
to which is a describing point, inside or outside the circum- 
ference of another base-circle. In the case of a rack, which 
may of course be looked upon as a wheel of infinitely large 
diameter, the base-circle is replaced by a straight line. 

66. Involute Teeth. — Taking first the case of involute 
teeth, let AR and BR (Fig. 127) be the pitch-circles of a 




Fig. 127. 

and h, a pair of wheels to be geared together witJi unitorm 
velocity ratio. R is the pitch-point. Now let MS and TN 
be a pair of circles concentric with the pitch-circles, and let 

PM ^PR 

QN QR' 

Draw MNy the common tangent to MS and TN ; evidently 
MN passes through R. Next suppose that MN represents a 
flexible string wrapped round the circles MS and TN and 
kept stretched between them. The desired velocity ratio 

(represented by the fraction ^j will correspond to the 



196 KINEMATICS OF MACHINES. 

rolling of BR on AR, and also to the relative movement f 
MS and TN, if connected by the string MN. Consider 
any point such as L, supposed to be fixed on the string. 
As the string unwraps from a, L will describe the curve SL 
on the wheel a, and SL will be an involute of the base-circle 
MS. Similarly, while the string is wrapping on to TN 
the point L will describe on the wheel h the involute LT of 
the base- circle TN . It is plain that the curves LT and LS 
must always touch at some point, L, on the line MN, which 
line is thus seen to be the path of the point of contact. Again, 
it is a property of the involute of a circle that the tangent, 
LN, drawn to the base-circle from any point L on the curve, 
is a normal to the curve at that point. The line MN is thus 
in every position of the wheels the common normal at the 
point of contact of the curves SL and LT, and it passes 
through the fixed pitch-point R. The condition for uniform 
velocity ratio is thus fulfilled, and if the teeth on a and h 
have their profiles formed on the curves SL, LT, they will 
work correctly together. The same curves would work 
together if the distance PQ were increased or diminished, 
for the common tangent MN would still divide PQ in the 
same ratio, and would still be the common normal at the 
point of contact. 

In order to complete the outlines of the wheels the num- 
bers of teeth must be decided. Evidently the length of 
tooth measured radially cannot be greater than UV, and 
must in practice be somewhat less. Further, in order that 
the wheels may work properly, a second pair of profiles 
must be commencing contact when the first pair cease to 
touch; in actual gearing at least two pairs of teeth are 
always in contact. This means that supposing the tooth 
profiles SL, LT are just ceasing to touch near A^, a second 
pair must be touching at R, and a third pair preparing to 
begin contact at or near M; in other words, RM must be 
not less than the distance from the point where the pitch- 
circle cuts the face of one tooth to the point where it cuts the 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 197 

face of the next. This distance, measured along the pitch- 
circle, is called the pitch, and the number of teeth on the 
wheel must be equal to the fraction (circumference of pitch- 
circle ^ pitch) . The pitch must plainly be the same for 
every wheel of a set gearing together, and we thus see that 
the numbers of teeth of wheels gearing together are propor- 
tional to the circumferences, or to the diameters, of their 
pitch-circles, 9.nd hence are inversely as their angular veloc- 
ities. 

The chief practical objection to the use of involute teeth 
is that the pressure between them acts obliquely along the 
line MN, instead of acting along a line perpendicular to V V, 
It is not possible to make involute wheels of only' a few 
teeth without increasing this obliquity to an undesirable 
extent. 

67. Cycloidal Teeth. — Cycloidal curves have the geomet- 
rical property that the normal to the curve at any point 
passes through the point of contact of the describing circle 
with the circle or straight line on which it is rolling. In 
Fig. 12% lei AR and BR represent the pitch-circles of a pair 
of wheels which are to have cycloidal teeth, P and Q being 
the centres of the wheels. As the pitch-circles roll together 
during the motion of the wheels, imagine that a third circle, 
CR, pivoted at Af , can roll in contact with the other two, 
and let L be a describing point on the circumference of CR, 
the three circles always touching at R. Suppose the three 
circles to move as shown by the arrows, and let 5 and T be 
the points on the pitch-circles of a and h respectively, which 
are in contact when L is at R. As the circle CR rolls on the 
outside of the circle BR, w^e may imagine that the describ- 
ing point L traces on the wheel h the curve TL, which is there- 
fore an epi'^ycioid. In the same way L traces on a the hypo- 
cycloid SL, and the curves SL and TL will, of course, always 
be in contact at the point L. Since R is the point on the 
circle CR which is at rest relatively to the circles AR and BR 
(for the circles roll and do not slip), it follows that the direc- 



198 



KINEMATICS OF MACHINES. 



tion in which L is moving at any instant relatively to the 
line of centres PQ, must be at right angles to the straight 
line LR, or, in other words, LR is the normal to the curves 
TL and SL at the point L. These cycloidal curves there- 




FiG. 128. 

fore fulfil the condition for uniform velocity ratio of the 
two wheels, and this fact does not depend on the size of the 
describing circle CR. 

The length and pitch of the teeth must be such that at 
least two pairs are always in contact, and the teeth are 
spaced out along the pitch-line exactly in the way described 
for involute teeth. Notice that the path of the point of 
contact for cycloidal teeth is an arc of the describing circle 
(arc Li? in Fig. 128). 

Cycloidal teeth in practice are almost invariably drawn 
by the use of two describing circles. Fig. 129 shows a pair 
of cycloidal teeth profiles just commencing contact at Lj, 
and just ceasing to touch at L^. The curves L^T^ and L^Sy^ 



CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING. 199 

are described by the circle C^R, while 1-2^2 ^^^ -^-2-^2 ^^^ 
drawn by a describing point on CJi. Note that the whole 
path of the point of contact is L^RL^. In the case of a 
single pair of wheels the circles C^R and C^R need not be 
of the same diameter, but if a set of wheels is to be made, 
any wheel of which is to gear with any other, the wheels 
must not only have teeth of the same pitch, but these teeth 




Fig. 129. 

(if cycloidal) must all be drawn with describing circles of 
the same diameter. 

Involute and cycloidal curves are, of course, not the only 
ones that can be chosen for the purpose of giving uniform 
velocity ratio. They are, however, easily drawn, and for 
this reason are generally used. 

We might have chosen arbitrarily a form for the teeth 



200 KINEMATICS OF MACHINES. 

of one wheel, and then, having given the centrodes (pitch- 
circles), we might have determined the proper form for the 
teeth of the second w^heel by the method of § 62. In some 
cases this would give a form impossible for construc- 
tive reasons, although correct so far as relative velocity is 
concerned. For instruction as to the design and propor- 
tion of wheel-teeth the reader should consult Unwin's 
' ' Machine Design," Vol. I, chapter X, or other works on the 
same subject. 



CHAPTER VIII. 

WHEEL-TRAINS ANJD MECHANISMS CONTAINING 

THEM. CAMS. 



68. Simple and Compound Wheel-trains. — The deter- 
mination of the velocity ratio in such a wheel-train as that of 
Fig. 130 involves no difficulty, for it is plain that one or 




Fig. 130. 

more intermediate wheels (as b) will not affect the numerical 
value of the velocity ratio of the first and last wheels. The 
linear velocity of the pitch-line of every wheel is the same^ 
and the angular velocity ratio of the first and last, there- 
fore, only depends on their own diameters, so that -^ = ± ~^, 



CO 



cd 



the sign depending on the number of idle wheels. Inter- 
mediate or idle wheels thus simply reverse the direction of 
motion. When all the wheels in the train have external 
contact, the angular velocity ratio of the first wheel to the 

last has a positive value (or, both wheels turn in the same 

201 



202 



KINEMATICS OF MACHINES. 



sense) if the number of axes is odd, while an even number of 
axes gives the velocity ratio a negative value. More com- 
plex wheel-trains, however, require further consideration. 
In Fig. 131 we have a compound spur-wheel mechanism of 




Fig. 131. 

four links, d being fixed, while h consists of two wheels rigidly 
connected and turning on the same axis. 

Let r^, r^, i^^,, r^, be the radii of the pitch-circles, then 
from § 64 we have 



ad 



0) 



bd 



Also, 



Hence 



ii) 



hd 



OJ 



cd 



OJ 



CO 



ad 
cd 






^aXRi 



= N, 



Suppose a to be the driving-wheel, while c is the driven 
one ; we see that the above result may be expressed by say- 
ing that 

revolutions of driving-wheel 



velocity ratio = 



revolutions of driven wheel 
product of radii of followers 



product of radii of drivers * 
Instead of radii we might evidently put numbers of teeth. 



WHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 203 

It would be easy to find a single pair of wheels having the 
same velocity ratio as the given train. For example, if we 
had a pair of wheels, A and C, such that 

^^-^c = r. + r, + i?, + r, and "f ^':^^\ 

these would have the same velocity ratio and the same dis- 
tance from centre to centre. The point of contact of their 
pitch-circles would divide the distance 0^^ 0^^ externally in 
the proportion of the angular velocities of a and c, and 
would in fact be the point 0^^.* Hence in Fig. 131 we 
have only to divide the line of centres, graphically or other- 
wise, in the proper ratio to find the sixth virtual centre. 




Fig. 132. 

In doing this (as in working all problems connected with 
wheel trains) , note must be taken of the sign of the velocity 
ratio, which depends on the presence or absence of annular 
wheels (i.e., wheels having internal contact), or of idle 
wheels, and also on the number of axes in the train. Take, 
for example, the two trains shown in Figs. 132 and 133, in 



* For graphic methods of determining virtual centres of wheel-trains, see 
Kennedy, Mech. of Machinery, note to Chapter VI. 



204 



KINEMATICS OF MACHINES, 



the first of which suppose r^ = 2, r^ = i, i^j== 2, 7^ = 1.5, so 
that the velocity ratio in Fig. 132 has the value 



N = 



(O 



ad 



CO 



- I ^-SX^ - I 3 



cd 



2X2 



8 



In Fig. 133 we have a simple train having exactly 
the same numerical value for its velocity ratio (since 

-^ = — ^ I , but in this case the negative value must be adopted, 

since the wheels a and c turn in opposite senses. In Fig. 132 
0^^ may be found by drawing A A' CC parallel to one another, 
and of lengths 8 and 3 respectively, to any convenient scale. 
The intersection of A'C with the line of centres fixes , 

ae 

Evidently the given train might be replaced by a pair of 
wheels of radii R^ and R^, the larger being annular, having 
their centres at A and C, and their pitch-circles touching 
at 0^^, as shown by the dotted arcs. Again, in Fig. 133 




Fig. 133. 

AA^ CC must be drawn parallel, but in opposite senses, so 
as to allow for the negative velocity ratio, and 0^^ is, of 
course, the point of intersection oiAC and A'C\ 

It is by no means necessary to have the centres of all 
the wheels of a train in one straight line. The back-gear 
of a lathe, for example, is an instance of a compound reverted 
train in which the centres of the first and last wheels coincide. 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 205 

This arrangement makes no difference in the numerical 
value of the velocity ratio, and is simply adopted for con- 
venience in construction. 

69. Epicyclic Gearing. — In the above examples of wheel- 
trains we have supposed the frame carrying the wheels to 
be the fixed link. Wheel gearing is often employed in which 
one of the wheels is the fixed link and the frame or arm 
carrying the remaining wheels is movable. Such gearing 
is called epicyclic, and we proceed to discuss some of its sim- 
pler cases. 

We take first the mechanism of Fig. 133, but suppose a 
to be fixed, while d is rotated in a clockwise or positive sense 
(Fig. 134). Let N be the velocity ratio of the train, i.e., let 



0) 



ad 




Fig. 134. 

Plainly, if we consider co^^ as being positive in sign, then cj^^ 
must be negative, hence 

(0^^= —CO. 



ad 



da' 



Now in any case where two bodies, c and d, have motion 
relatively to a third, a, which is fixed, any angular move- 
ment of c relatively to a may be looked on as the algebraic 



2o6 



KINEMATICS OF MACHINES. 



sum of the motions of c relatively to d and of d relatively 
to a. Thus 



= — (1) 




or 



OJ 



-^ = i-N, 



0) 



da 



where N is itself a negative quantity. A numerical example 
may, perhaps, make this clearer. Suppose the wheels a and 
c to have loo teeth and 90 teeth respectively; these teeth 
have the same pitch, and we can, of course, take the ratio of 
the numbers of teeth instead of the ratio of the diameters or 

T 10 

radii of the pitch-circles. Thus — "" = — ; m other words, 

fc 9 
supposing d to be fixed, while a makes one revolution with 




Fig, 135. 
regard to d, c would make i| in the opposite sense. Now 
suppose that in a certain time a makes — i revolution, c 
making -\-\\, while d is at rest. Cause the whole mechan- 
ism to execute + 1 rotation in the same time around 



ad > 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 207 



this brings a to rest, makes d perform + 1 revolution, and 
therefore gives c 1^+1=2^ revolutions in the same sense as 
that of the arm. 

If an idle wheel, b, had been interposed between a and c, 



CO 



cd 



as in Fig. 135, we should have had A/ = +-- =~^ = aposi- 



OJ 



cd 



tive quantity, and ^^^ = ^d a + ^ccf whence -^ -= i + — ^'^ =' 

I — A^ as before. With the numbers of teeth, as in the 
example just given, and the train arranged as in Fig. 135, 
we should have, if A^ = + i-l, 



CO 



-ii = -i; 



da 



i.e., for each revolution of the arm, c makes i revolution in 
the reverse sense. 

Fig. 136 represents a compound epicyclic reverted train. 
Let n , n, n. n. he the numbers of teeth in the wheels 




Fig. 136, 

a, 61, b2, and c respectively. Evidently, if the pitch of both 
pairs is the same, n^ + n^ =n^ +^,- The velocity ratio of 
the train will be positive and has the value 



N 



(O 



cd 



n\n. 



CO 



ad n^^Xn^ 



208 



KINEMATICS OF MACHINES. 



hence 



CO 



cd 



^-{NXcoJ. 



Further, when a is the fixed Hnk 



(o 



ca=^.d+^da=^da(l-^)' 



Thus, for instance, suppose a has 30 teeth, 5^ has 15, and 
62 and c have respectively 20 and 25 ; then 

CO. 



15X25 



_ ""cd 



CO 



ad 



and 



CO. 



« =i-N=-o.6. 



CO 



da 



Thus c will make 0.6 revolution for each revolution of the 
arm, but in the opposite sense. Such a train might evi- 




FiG. 137. 

dently be arranged to give c a very slow rotary motion, say 

revolution for each revolution of the arm d. 

10,000 

As an example of an epicyclic reverted gear containing 
an annular wheel the wheel-train used in certain front- 
driving bicycles * maybe given. In Fig. 137 // represents 



* See also the Weston triplex pulley-block described in § 78, Chapter IX. 



iVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 209 

part of the front fork of the bicycle, to which is rigidly 
attached the central pinion /'. The arms a, secured to 
the axle and cranks, carry one or more planet- wheels, p, 
gearing with the central pinion and with an annular wheel 
formed on the inside of the hub, h, of the driving-wheel. 
Suppose this wheel h has 60 teeth, while p has 1 5 and /' has 
30; it is plain that %= n^-f-2n^ if the wheels are to gear 
together and the wheel h is to be coaxial with /'. 
Now if a were the fixed link. 



— _^ — _i. 

therefore (o,^^=-^X (o^^ = iw^jr. 

The ratio to be determined is the number of revolutions of 
the wheel h per revolution of the crank a ; this is the same 
quantity as 

— ^ = velocity ratio of h and a. 

Now OJ^j:--=OJ,,^-\-OJ^f 

= (4 + i)^./. 
Hence ^^=-1-1.5; 

in other words, the wheel will make i J revolutions for each 
revolution of the crank, and in the same sense. A bicycle 
having a driving-wheel 44 inches diameter would therefore 
be geared to 66 inches with this arrangement. 

70. Mechanisms Containing Wheel-trains. — Mechanisms 
are of common occurrence in which wheel trains form part 
of chains containing also sliding and turning pairs. Fig. 138 
shows diagrammatically a * * sun-and-planet ' ' gear contain- 
ing an annular wheel, forming part of a mechanism con- 
taining a slider-crank chain. 

The crank a is able to rotate about the point 0^^ with 
reference to a fixed frame d, and pairs with a link b, forming 
the connecting-rod in a slider-crank chain, of which c is the 



2IO 



KINEMATICS OF MACHINES. 



sliding block. On h, however, is formed a spur-wheel 
whose pitch-circle has its centre at 0.,^. The spur-wheel 
gears with an annular wheel e whose pitch -circle has its 
centre at 0^^. The virtual centres are marked on the dia- 




FiG. 138. 

gram. We wish to find the number of revolutions of the 
annular wheel e for each revolution of the crank a. 

As in previous examples, let N be the velocity ratio of 
the wheel-train ; i.e., let 



iV = 



'J). 



(O 



ba 



^ 

n. 



Plainly N will be a positive fraction in this case. We note 
that during the action of the mechanism the average value 
of ojj^^ is zero, for b simply swings to and fro, and a line 
marked on it describes equal angles right and left from its 
mid-position. Hence we may say that on the average 



(O 



ab 



= (0. 



CO 



ad ' 



'ad + ^db 

that is, we may consider the angular velocity of a and b 
instead of that of a and d. The two would always be exactly 



IVHEEL-TRAIhlS AND MECHANISMS CONTAINING THEM. 211 



equal if h always remained parallel to itself, i.e., if the con- 
necting-rod were infinitely long. 

Now ^.^ = ^.^ + ^. 



CO 



ed 



ad ' ^a-' 
. CO.. 



CO 



ad 



CO 



ad 



CO, 



= 1 



= I 



(J) 



da 



CO^ 



CO 



ba 



= i-N. 

For example, suppose e had 100 teeth while b had 95, so that 
N = +0.95 ; then for each revolution of the crank a, e would 
make 1 — 0.95=0.05 revolution in the same sense. This 
mechanism is actually used as gearing for a capstan driven 
by a hydraulic engine, b being attached to the connecting- 
rod, while the capstan barrel is attached to e. 

As another example of a mechanism containing a wheel- 
train we may take the wheel crank- chain of Fig. 139, which 




Fig. 139, 

is formed by combining a simple wheel chain with an open 
crank-chain of five links. 

If the lengths of the links a and b, and also those of c 



2 12 



KINEMATICS OF MACHINES. 



and d, are equal, as in the figure, we obtain Cartwright's 
straight-line motion, in which the point P describes a 
straight-line path passing through Q. The purpose of the 
two spur-wheels is to close the five-link chain, whose motion 
would otherwise be unconstrained. 

In Figs. 140a and 1406 we have a slider-crank chain in 
which spur-wheels can be used for a somewhat similar pur- 
pose. Consider a slider-crank chain in which the connect- 




FiG. 140a. 




CENTRODE OF 6 

(d fixed) 



Fig. 1403. 

ing-rod h is made equal in length to the crank a. (Compare 
Figs. 60 and 91.) With these proportions it is possible for 
the stroke of ^ to be (i) either twice the length of the crank,. 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 213 

as in the ordinary slider-crank chain, or (2) four times the 
length of the crank, in which case 0^^ must travel on past 
0^^. A third possibility is that 0^^ and 0^^ may remain 
coincident, in which case h and a move together, and the 
mechanism will reduce to a pair of elements. 

Notice in Fig. 140a that since the length of h is equal to 
the length of a and the angle O^jD^fi^^ is aright angle, the 
points O^jD^jO^j^ must lie on the circumference of a circle 
whose radius is the length of a or the length of h. Since 
O^JD^^is a diameter of this circle, it follows that 0^,^ remains 
always at the same distance from 0^^, and the centrode of 
h with regard to c? is a larger circle whose radius is twice the 
length of a. If now (Fig. 1406) we attach to d an annular 
wheel whose pitch-circle is the centrode of h with regard to 
d, and if we fix to 6 a spur-wheel whose pitch-circle is the 
centrode of d with regard to h, these wheels will gear 
together, and will compel 0^^ to remain always at a fixed 
distance from 0^^. If these wheels were not provided we 
should have a change -point at the instant when 0^^ passes 
0^^, but if the virtual centre 0^^ is compelled to remain at 
a fixed distance from 0^^ by the action of the spur-wheels, 
0^^ is compelled to continue its travel, and the mechanism 
is not permitted to change. Obviously this arrangement 
is really a case of pair-closure at a change-point. (Com- 
pare the examples in § 59.) The only really essential 
portions of the wheels are therefore those teeth which are 
in gear while 0^^ is passing 0^^. 

71. Cam- trains. — The name cam- train is applied to 
mechanisms containing a rotating disc (generally non-circu- 
lar) or a sliding plate, the profile of which forms one element 
of a higher pair and gives some desired periodic motion to 
the second element of the pair. Such a cam-pair may be 
closed by forming one element into the geometrical envelope 
of all possible positions of the other element. Mechanically 
cam-pairs usually possess the disadvantage of small wearing 
surface and rapid wear, common in higher pairs. Almost 
invariably force-closure is necessary to make up for the 



214 



KINEMATICS OF MACHINES. 



looseness of fit following on wear. A cam-train is in general 
a mechanism of three links ; such, for example, is the cam- 
train found in the stamp-mill used for crushing hard ores 
(Fig. 141). 




Fig. 141. 



A rotating shaft carries the cams hh. These successively 
lift and let fall the stamps cc, which are guided by means of 
the framework a. It will be noted that the cam-pair he is 
force-closed by the weight of the stamp itself, and also that 
the form of the cam is such that in any position during the 
upward stroke it is touched by the horizontal under surface 
of the collar on the stamp-rod. This fact has to be consid- 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 215 

-ered in selecting the form of the cam, for it is obvious that 
during all the upward movement the point of the cam sur- 
face which touches the collar must be at a higher level than 
any other point on the cam surface. It will be found that 
with such a cam it is not possible to give the collar any 
arbitrary position for any given angular position of the 
•cam.* 

In many cases the cam has to determine the position of 
a point, instead of a flat surface, on the follower or link driven 
by the cam. This point is then usually taken as the centre 
of a roller or pin with which the cam engages; and within 
certain limits imposed by constructive considerations, any 
desired continuous change of position can be given to the 
follower by suitably choosing the form of the cam profile. 

72. Rotating Cams. — The action of cam-trains will be 
most easily understood by the study of a few examples. To 
produce a given form of reciprocating motion along a straight 
or curved line we may employ either a rotating, a sliding, or a 
cylindrical cam. The first example (Fig. 142) will be that of 
a rotating cam designed to give its follower a reciprocating 
motion along a straight line passing through the cam centre, 
the velocity being uniform throughout both strokes if the 
cam rotates with uniform angular velocity. The mechan- 
ism is somewhat similar to that of Fig. 141, and consists of 
a cam c (whose form is to be determined), a guiding frame 
a, and a follower h, which is to slide with the periodic 
motion specified above. The end of the follower is pro- 
vided with a roller, for the sake of lessening friction. 

Since the cam rotates uniformly, while the follower moves 
with uniform velocity, the cam describes equal angles while 
the follower traverses equal distances. Plainly the outline 
of the curve required will be such that vSuccessive radii 
making equal angles with one another have a constant 
difference in length. All we have to do, in fact, is to divide 
the path of A into any convenient number of equal parts, 

* See Kennedy, Mech. of Machinery, p. 154. 



2l6 



KINEMATICS OF MACHINES. 



say six, and to divide the half-revoltition of the cam into 
the same number of equal angles. If CA^ is the least dis- 
tance of the centre of the roller from the centre of the cam, 
and CA^, CA^, etc., are the distances after one, two, etc., 
twelfths of a revolution, we then make Ca^ = CA^, Ca^ = CA^y 
and so on. The curve drawn through a^a^ will be recog- 




FiG. 142. 

nized as an Archimedean spiral, and the same kind of curve 
will, of course, be found for the remaining half of the cam. 
In this curve let CA ^ = r^, and let r be any radius-vector of 
the curve, then r = r^ + md, where 6 is the angle the radius- 
vector makes with CB and w is a constant. The real out- 
line of the cam itself is not the dotted line a^a^a^, but the full 
line drawn so as to touch a series of circles whose centres 
lie on a^a^, etc., and whose diameters are all equal to that 
of the roller on the follower. 

A cam frequently has to actuate a point on a lever, 
which of course moves in the arc of a circle. An exactly 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 217 

similar construction in this case gives the form of the cam, 
but the points .4 Q, A^, ... , will now be placed along a cir- 
cular path instead of along a straight line. 

As a more difficult case, let us consider the form to be 
given to a cam arranged to move a follower with uniform 
acceleration during one half of a revolution, after which the 




Fig. 143. 

sliding piece remains at rest during a quarter of a revolu- 
tion and returns with uniform velocity during the remain- 
ing quarter revolution, the cam rotating uniformly. 

In Fig. 143 let the points a, b, c, d, e correspond to the 
positions of the centre of the pin on the reciprocating piece 



2l8 



KINEMATICS OF MACHINES. 



at equal intervals of time during one half revolution, while 
the distance ae is the length of stroke of the reciprocating 
piece. Since the upward stroke is to be made with uniform 
acceleration, the distance ac = 4ab, while ad = gab, and ae 
= i6ab (where ab is the distance moved in one eighth of a 
revolution). Let be the centre of the cam. Then, start- 
ing with the sliding-piece in its lowest position, when the 
cam has turned through a quarter of a revolution, the line 




Fig. 144. 

Oc' will coincide with Oc. Thus at the instant when the 
sliding -pin has its centre at c, the radius Oc' will be vertical, 
and we must set off a distance Oc' = Oc. By a similar con- 
struction other points on the curve, such as b\ d\ etc., are 
obtained. The profile of the cam itself is obtained by 
drawing a curve (shown in full lines) at a uniform distance 
from ab'c'd'e' equal to the radius of the pin on the sliding- 
piece. 

We may now consider the construction to be adopted if 
the line of motion of the follower-point does not pass through 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 219 



the centre of rotation of the cam. In Fig. 144, let 
ylo^iAj... be successive positions of the follower-point, 
corresponding to successive equal angles described by the 
cam-shaft, and let the line A^A^A^ . . . produced be a tangent 
to a small circle BqB^B2B^ . . . described about the cam centre. 

As the cam rotates it is seen that such a line as Effi^^ 
drawn touching the small circle will take up the position 
5 qA 0^6 when it becomes vertical. Hence the point a^ will 
be found by drawing a circle with as centre, and radius 
OAq so as to cut B^a^', then B^a^ is the line which coincides 
with B^A^dit the time when the follower-point is at A^. 

If the distance A^A^ is equal to the arc B^fi^, and the 
distances A^A^, B^B^, dire equal, and so on, it is evident that 
the curve a^a^ is an involute of thebase-circle^^^i^a-^a* • • 
Such curves are generally used for the cams of an ore- 
crushing stamp-mill. In an involute the tangent to the 
base-circle is a normal to the curve (see § 61); hence in 
any position of an involute cam the point lying on the 
vertical line B^A^ will touch a horizontal line corresponding 
to the under surface of a collar on the stamp-rod. 

73. Sliding and Cylindrical Cams. — The form of a slid- 




Fig. 145. 

ing cam to obtain any desired kind of periodic motion is 
easily determined. Fig. 145 shows the arrangement of a 
cam of this kind used for giving the requisite motion to the 



220 



KINEMATICS OF MACHINES. 



belt-shifting gear of a planing- machine. The cam c here 
takes the form of a slotted plate, sliding in a frame or guide 
a. Two bell-crank levers, h^ and h^, pivoted to a at L and 
M, carry follower-pins which work on the slot in c. The 
longer arms of the levers are provided with forks for shifting 
the belts as required from the fast to the loose pulleys of 
the planing-machine. The length and form of the slot in 
the cam are such that when c is in its extreme position to 
the left, as shown by dotted lines, h^ is thrown to the right, 
and its belt runs on the fast pulley P^, while the fork h^ is 
also inclined to the left and its belt runs on a loose pulley P3. 
On moving the cam from left to right it will be seen that 
^2 first moves its belt on to the loose pulley P^, and after- 
wards 6j moves the second belt on to the fast pulley P^. In 
this way it is impossible for both belts to be on fast pulleys 
at the same time. The belts drive the pulleys P^ and P^ 
in opposite senses, hence the action of the gear is to reverse 
the motion of the shaft to which the pulleys are attached. 
The motion of the planing-machine table is thus also re- 
versed. 

Fig. 146 shows a cylindrical cam, in which the cam profile 




ZI 



Fig. 146. 



is traced on the surface of a rotating cylinder and the line of 
motion of the follower is parallel to the axis of the cylinder. 
The figure shows how such cams are employed in certain 



IVHEF.L-TRAINS AND MECHANISMS CONTAINING THEM. 221 

automatic screw-making machines for the purpose of giving 
endwise motion to the rotating spindles carrying the work. 
The working profile of the cam is the edge of a strip secured 
to the surface of the cylinder c by screws ; a series of these 
strips may evidently be arranged so as to give any desired 
periodic range of rest and motion to the carriage of the 
rotating spindle. In this case the roller on the follower h 
is kept pressed against the cam edge by a spring or other 




Fig. 147. 

suitable means. The mechanism is thus force-closed. It 
would, however, be quite easy, by attaching two strips, to 
form a groove in which the follower-roller would work ; the 
mechanism would then hsive pair-closure. 

A rotating cam of the kind shown in Fig. 143 could be 
closed in the same manner if a groove were formed on the 
flat surface of the cam-plate, engaging with a pin or roller 
attached to the follower. Or, as an alternative, the cam 
may be formed with a figure of constant breadth, in which 



222 



KINEMATICS OF MACHINES. 



case it can have constrained motion with relation to a fol- 
lower provided with a pair of parallel faces, as in Fig. 147. 

Cams which are pair-closed are often called positive- 
motion cams. 

74. Velocity Ratio in Cam- trains. — Let Fig. 148 repre- 
sent a cam- train of three links ; we wish to find the velocity 
ratio of the train, i.e., the ratio 

linear velocity of follower 
angular velocity of cam 




Fig. 148. 

Find 0^^, the centre about which the cam rotates, and 
also 0^^, which in this case is at an infinite distance, since 
the follower moves in a straight line, QP. If the follower 
were a lever turning about a fixed centre, the line O^fl^^ 



IVHEEL-TRAINS AND MECHANISMS CONTAINING THEM. 223 

would be as easily determined, The third virtual centre 
0^^ must be the point where the common normal to the cam 
and follower at their point of contact cuts this line. Let 




Fig. 149a. 




Fig. 1493. 

V be the common velocity of the bodies h and c at their 
point of contact ; its direction will be parallel to the line MP. 
Draw a triangle of velocities ABC, in which AB represents 
F, CB represents V^, and AC represents the velocity of 



224 



KINEMATICS OF MACHINES. 



sliding of c on 6 in a direction parallel to the common tan- 
gent at the point of contact. Draw O^JVL parallel to this 
tangent, and therefore perpendicular to PM, 




Fig. I49<r. 




Fig. 149^. 

Then the angular velocity of the cam 



V 



ca 



O.M 



and the linear velocity of the follower along QP 



IVHEEL TRAINS AND MECHANISMS CONTAINING THEM. 225 
Hence the velocity ratio of the pair is 



F, 



OJ. 



v-o^p,, o^M 



O.M 



V 



O^fi^. 




Fig. I49<f. 




Fig. 149/. 

In other words, if the cam rotates with uniform velocity, the 
linear velocity of the follower is proportional to the distance 
between the centre of rotation of the cam and the virtual 



2 26 KINEMATICS OF MACHINES. 

centre of the cam and follower. If the point P moves in an 
arc of a circle about a point 0^^ as in Fig. 1496, then it 
may be shown, exactly as in § 63, that the angular velocity 
ratio of h and c will be the number 



A number of the forms taken by cam-trains are illus- 
trated by the models represented in Figs, i/^ga-i^gf. A 
positive-motion cam, of form somewhat similar to that of 
Fig. 142, is shown in Fig. 149a, while Fig. 1496 is an exam- 
ple of a cycloidal cam whose follower point moves in a 
circular arc. Fig. 149c shows another positive-motion cam, 
where the follower-pin works in a groove in the cam, and 
Fig. i^gd is a sliding cam. In Fig. 149^ we have a rotat- 
ing globoidal cam actuating a lever; Fig, 149/ represents 
the form of cylindrical cam known as a ** swash-plate." 
The reader will notice in three of these cases the springs 
which close the pair. 



CHAPTER IX. 

RATCHET MECHANISMS AND ESCAPEMENTS. 

75. Ratchet-gearing. — We have so far considered mech- 
anisms in which relative motion of the various links is possi- 
ble at any instant, so that no link is definitely held or checked 
by another. We have now to study the action of Ratchet- 
gearing, which may be said to be gearing so arranged that 
certain links are temporarily or periodically locked together 
or connected during the action of the mechanism. This 
locking or checking of relative motion may be so effected 
that relative motion of the two links is only possible in one 
sense or direction (when the gear is called by Reuleaux a 
Running-ratchet Train), or movement in both directions 
may be rendered impossible when the ratchet acts, in which 
case the gear is known as a Stationary-ratchet Train. Fig. 
150 shows the two kinds of ratchet-train in their typical 





Running. Fig. 150. Stationary. 

forms. Each consists of a frame or arm a, a ratchet-wheel b, 

and a ratchet or click c. In the first figure b is evidently 

capable of left-handed rotation only, so long as the ratchet c 

(sometimes called a pawl) is resting against its teeth. In 

the second figure motion is only possible when the pawl is 

227 



228 



KINEMATICS OF MACHINES. 



lifted clear. Examples of simple ratchet-trains will readily 
occur to the reader; in Fig. 151, for instance, is shown the 




Fig. 151. 

mechanism of a ratchet-drill, in which the different links are 
lettered in the same way as in the preceding figure. 

76. Running Ratchets. — It is not necessary that the con- 
nection between the pawl and ratchet-wheel in a running 
ratchet should be of the positive kind shown above. Fig. 
152 shows a form of frictional ratchet gear commonly used 
to transmit motion in one sense only from the crank-axle 
to the sprocket-wheel of a " free-wheel" bicycle. Here the 
ratchets themselves, cc, take the form of small rollers held up 



RATCHET MECHANISMS AND ESCAPEMENTS. 



229 



by springs behind them; the rollers are confined within a 
driving-ring, h, attached to the sprocket-wheel, and when in 
action jam between this ring and suitably formed surfaces 
on a ratchet-wheel, a, attached to the crank-axle. Such 




Fig. 152. 

frictional ratchet gears are sometimes classed under the 
head of silent ratchets. 

It should be noted that while ratchet- trains are used most 
frequently for controlling the motion of a turning pair, 
there are many cases in which such trains actuate links 
which have linear motion. 

Fig. 153 shows a running-ratchet gear in which the 
ratchet c, attached to a reciprocating bar d, acts on a 
ratchet-rack h, and drives it in one direction only, motion 
in the opposite direction being prevented by a second ratchet 
or pawl c' , attached to the fixed link a. The mechanism is 
thus a combination of two running-ratchet trains, ahcd and 
ahc' ; the former for driving, the latter for checking. 



230 KINEMATICS OF MACHINES. 

Most running ratchets in common use are really a com- 
bination of this kind; for example, in the ratchet-drill the 
function of the checking ratchet is performed by the fric- 
tional resistance of the drill in its hole. 

It is important to note that the form of the surfaces on 
which the pawl and ratchet-wheel or rack engage must be 
carefully chosen, in order that .the mechanism may fulfil 
its purpose. The shape of the pawl must in fact be such 
that the pressure between it and the tooth or surface with 
which it acts does not tend to throw it out of gear. Further, 
the mechanism must be force-closed, so that the pawl always 
tends to engage itself; this is commonly effected either by 
the action of springs (Figs. 151 and 152), or by the weight 
of the pawl itself (Figs. 150 and 153), or, in some cases, by 
making the pawl itself a spring. Fig. 154 shows a running 
friction ratchet which depends for its action on the weight 
of the ring-shaped pawl itself. Such a mechanism has been 
employed in certain electric arc lamps for controlling the 
downward movement of the carbons. 

77. Statioifary (Checking and Releasing) Ratchets. — 
Ratchet mechanisms of this type are used where it is 
necessary to check and release the driven link at will. In 
most cases a running ratchet or a cam is provided for the 
purpose of actuating the link whose motion is controlled by 
the locking ratchet. The mechanism of a lever-lock (shown 
diagrammatically in Fig. 155) is of this kind. The tumbler c 
and the bolt h here form a stationary-ratchet mechanism 
with the frame a. 

The release of the bolt is effected by the action of the 
portion M of the key, which really forms a cam engaging with 
the curved surface of the profile PQ of the tumbler. When 
this release has been effected the bolt is shot back by the 
action of the portion N of the key. This part (also a cam) 
moves the bolt by engaging with the notch seen on the under 
side of the bolt. In actual lever-locks three, four, or more 
tumblers are used, with a corresponding number of steps on 



RATCHET MECHANISMS AND ESCAPEMENTS. 231 





Fig. 153. 



Fig. 154- 




Fig. 155. 



232 



KINEMATICS OF MACHINES. 



the key, and springs are provided so as to press the tumblers 
against the key. 

Releasing and checking ratchets need not necessarily be 
positive in their action ; they may depend on f rictional forces 
just as in the case of the driving ratchet of Fig. 152. Thus, 
for example, a friction-brake may be looked upon as a fric- 
tional checking ratchet. 

In Figs. 156a and 1566 we have another example of a 
checking-ratchet train, in the case of the Yale lock. This 




Fig. 156a. 




Fig. 156-5. 
lock really contains two distinct mechanisms, one a cam- 
train ahc, which actuates the bolt, and the other a locking- 
ratchet train, which secures the cam, and can only be re- 
leased by the insertion of the correct form of key. These 
mechanisms are shown separately. Fig. 156a shows the 



RATCHET MECHANISMS AND ESCAPEMENTS. 



233 



former train, in which the cam c is rotated by turning the 
key, and locks the bolt when in its extreme outer position. 
Fig 1566 shows the cara and its bearing; on inserting the 
notched key /, as shown, each of the tumblers or pawls e is 
lifted to such a height that the division between the two 
portions of the tumbler is flush with the surface of the 
bearing. The cam can then be rotated and the bolt h 
can be shot or withdrawn. This locking gear is, of course, 




Fig. 157a. 
a stationary-ratchet train. The case a^ is rigidly connected 
with the frame of the lock, a, when the whole lock is put 
together. 

Most checking or releasing ratchets are found combined 
with some form of cam gear, as in the examples above. This 
is also shown in the case of the releasing-ratchet trains em- 
ployed for working the steam- valves of a Corliss engine (Figs. 
157a and 1576). Fig. 157a represents the engine cylinder 
and the gear for working its steam- and exhaust -valves ; 
Fig. 1576 shows in diagrammatic form the ratchet mechan- 
ism of the steam- valves. The various parts are arranged 
somewhat differently in the two figures. The object of such 
gear is to open the valve at the proper point in the revolu- 
tion of the engine, and then, after a variable interval, de- 



234 



KINEMATICS OF MACHINES. 



pending on the amount of steam to be admitted, to release 
the valve so that it may be promptly closed by the action of 
springs or of gravity. The valve is attached to the spindle 
and lever a, and is opened by rotation in the sense shown by 
the arrow. 




Fig. i57<5. 
In the example shown, the point at which the valve 
closes is determined by a cam d whose position is regulated 
by the governor of the engine. During the motion of open- 
ing, the valve is driven from the rod / connected to a rock- 
ing wrist-plate (Fig. 157a). The wrist-plate thus gives a 



RATCHET MECHANISMS AND ESCAPEMENTS. 



235 



rocking motion to the lever e, and when moving in the direc- 
tion of the arrow this lever opens the steam- valve by the 
engagement of the ratchet or pawl c with a corresponding 
stud or projection on a. On reaching the proper point the 
pawl is lifted by the action of the cam d ; then the weight 
of the dashpot, or the tension of a spring, causes the lever 
a to drop. Thus the valve is promptly closed. A spring 
(not shown) is of course required in order to keep the pawl 
c pressed against the cam d and in readiness to engage with a 
on the return stroke. 

The many forms of brakes and clutches may be regarded 
in a sense as ratchet mechanisms (checking and releasing 
ratchets) ; in many cases their action is independent of the 
sense in which the wheel or shaft is rotating. 

Fig. 158 shows two forms of clutch employed for connect- 
ing at will two pieces of shafting, A and B, To the shaft 
B is secured one portion of the clutch 5^ ; the shaft A carries 




{ 






Ai 



:£l 



B 



i 



Fig. 158. 

the other portion, A^, in such a fashion that A^ may be made 
to slide along A so that its projections will engage with the 
corresponding recesses in B^. At the same time the pro- 
jecting feather or key A^ compels A^ and A to rotate to- 
gether. Thus when the clutch is engaged, the rotary motion 



236 



KINEMATICS OF MACHINERY. 



of B is necessarily transmitted to A. On comparing Figs. 
158 and 150 the reader will see at once that we have in the 
two forms of clutch an exact equivalent of the running and 
stationary ratchet of § 75. The clutch shown in Fig. 158 
in the upper view will only transmit relative motion in one 
sense ; it is therefore really a running-ratchet gear. In the 
lower view no relative movement of the shafts is possible 
when the clutch is engaged; the contrivance thus forms a 
stationary ratchet. 

An example of a frictional running ratchet was given in 
§ 76. Fig. 159a represents a locomotive wheel and its 
brake; here we have essentially a frictional stationary- 
ratchet gear used as a brake, the brake-block corresponding 
to the ratchet or click. In Fig. 1596 we have a frictional 




Fig. 159a. 




Fig. 159^. 



RATCHET MECHANISMS AND ESCAPEMENTS, 237 

stationary ratchet used as a clutch for communicating 
motion from the shaft B to the shaft A . When the clutch 
A^ is pressed along the shaft into contact with B^ the fric- 
tional grip between the two halves of the clutch is sufficient 
to drive the shaft A. The half clutch A^ is made to slide 
along the shaft by the action of a fork whose jaws engage 
in the groove G shown in the sketch. The same arrange- 
ment is employed in the clutches of Fig. 158. 

Ratchet mechanisms are of very frequent occurrence 
in machinery, and it is here impossible to attempt any ex- 
haustive catalogue of their many forms. The subject has 
been most completely treated by Reuleaux.* Certain 
ratchet mechanisms containing non-rigid links are discussed 
in §88. 

78. Escapements (Uniform, Periodical, and Variable). — 
Under the head of escapements may be classed a number of 
self-acting checking and releasing ratchet mechanisms in 
which the driven link is alternately released and stopped. 
The most familiar example is, of course, found in a clock or 
watch, where the driving weight or spring is permitted to 
move the clock-work and the hands by a definite amount at 
regular intervals. Such an escapement is a uniform escape- 
ment. A second kind of escapement (e.g., the striking 
mechanism of a clock) allows a train of wheel-work to move 
at definite intervals, but the amount or range of movement 
is varied in a predetermined manner, so that, for instance, 
at every hour the striking gear makes one more stroke than 
at the preceding hour, up to twelve strokes, after which the 
cycle commences again. We have here a periodical escape- 
ment in which, while the period is constant, the range is 
periodically variable. There is still a third kind, an adjust- 
able escapement, in which the range or the period is variable 
at will or is altered irregularly. We shall take an example 
of each kind. 



* The Constructor, Chapter XVIII ; Kinematics of Machinery, §§ 119-121. 



238 



KINEMATICS OF MACHINES, 



Graham's Escapement (Fig. i6o) belongs to the first class, 
and its essential parts are an escape-wheel a (connected with 
the wheel- work of the clock and driven by it in the sense 




Fig. i6o. 

shown by the arrow), and an anchor 6, whose motion is 
controlled by the pendulum, with which it is connected 
through the verge and fork U . The escapement must (i) 
permit the escape-wheel to advance by one tooth at each 
swing of the pendulum, and must also (2) communicate at 
each swing a minute impulse to the pendulum, so as to 
maintain its periodic motion. The anchor is really a ratchet, 



RATCHET MECHANISMS AND ESCAPEMENTS. 239 

the surfaces LM, L'M' forming the working faces of the 
pawl when the motion of the escape- wheel is checked. The 
faces LN , L'N' are slightly inclined to the circle passing 
through the tips of the teeth of the escape-wheel, so that as 
each tooth is driven past the pallet or point of the anchor, a 
small impulse is given to the pendulum while near the centre 
of its swing. Almost immediately after a certain tooth has 
passed LN , for instance, the anchor swings from right to left, 
and the escape- wheel is checked, because another tooth 
strikes the face UM' only to be released when the pendulum 
again swings back. The curved portions of the tooth-out- 
lines are so formed as to clear the points of the pallets while 
the anchor is receiving its impulse. It is important that a 
good clock escapement should work well with a very small 
angular movement of the pendulum; aiid in this respect 
Graham's escapement was a great advance on its predeces- 
sors. 

As an example of a periodical escapement we may take 
the so-called "English" striking- train of a clock. Fig. 161 
shows this mechanism in a diagrammatic form, omitting all 
unnecessary details. It is desired to communicate to the 
hammer of a bell or gong such a periodic motion that at 
stated intervals, say of one hour, the bell is struck; the 
number of strokes increasing by one each time the move- 
ment occurs until the cycle is completed. The whole con- 
trivance includes 

(i) A train of wheels (c^gjk) set in motion by its own driv- 
ing weight or spring and checked by a ratchet which is 
released every hour by the clock itself. 

(2) A mechanism (driven by the clock) which controls 
the range of movement of the wheel train c^gjk, and thus 
varies the number of strokes given by the bell. 

The first part of the escapement consists of the wheel 
c^, the cam c^, and the single -toothed wheel c, all rigidly con- 
nected ; gearing with c^ is the wheel g, provided with a pin 
on which the pawl /^ acts ; and gearing with g is the wheel 



240 



KINEMATICS OF MACHINES. 



k, carrying a number of pins which move the hammer of the 
bell as k rotates. The whole of this gearing is driven in the 
sense shown by the arrows and is not directly connected 
with the driving mechanism of the clock itself. Its move- 
ment can only take place when the pawl /^ is dropped so 
as to clear the pin on g. There is, however, another way 
of checking the motion of c^, g, j, and k. Suppose that the 
pawl /o is released and that c^ moves in the sense of the arrow 




Fig. 161. 

from the position shown in the sketch. This motion will 
continue, and the single tooth of c will engage with the teeth 
on h until that sector has been lifted to its highest position 
when c clears the teeth on the sector, and comes in contact 
with the stop at the lower corner of 6. It will be seen that 
the cam c^ performs another office, for it lifts the pawl a, 
and releases h, every time that the tooth on c is in gear, 
and it also permits this pawl to drop and hold the sector b 
during the time that the tooth on c is not in gear. The 



RATCHET MECHANISMS AND ESCAPEMENTS. 241 

sector, the wheel and cam, and the pawl thus form a sepa- 
rate locking and releasing ratchet-train and act soniewhat 
after the fashion of the train of § 7 7 . 

The upper position of the sector is definite; the lower 
position evidently depends on the position of the "snail" 
e. If this snail is driven by the clock in such a fashion that 
it advances one division every hour, it is evident that the 
range of the sector h will be altered every hour also. It 
only remains to arrange that a spring h shall tend to make 
the sector assume its lowest position, and that a pin on a 
shall be lifted by f^ so as to release the pawl a and allow 
the sector to drop whenever the movement of the train c^gjk 
is prevented by Z^. " 

The action of the whole escapement is then as follows: 
When the wheel d (which is geared with the snail e) ad- 
vances beyond the position shown, /^ drops and permits the 
train c^gjk to be set in motion. The bell is: then struck as 
many times as is permitted by the range of the sector 6, 
and that sector is left iii its upper position; the motion of 
6, c, g, and k then ceases. When d has raade another revo- 
lution and the snail has advanced one division, f^ is again 
lifted, the pawl a is also raised, and the sector at once 
drops, ready for the train to strike again as soon as g is 
released by the further movement of the wheel d. 

In an actual striking- train there will, of course, be twelve 
divisions on the snail and a corresponding number of pins 
on k. 

The examples here described will serve to give some 
idea of the nature of escapements of the first two kinds. 

Adjustable or variable escapements form a most important 
class of mechanisms from a practical point of view; they 
are often of considerable complexity. For instance, the 
steering-wheel, steering-engine, and rudder of a ship form 
together a complex variable escapement. This will be 
understood when it is pointed out that it is necessary for 
the rudder to move through an angle exactly proportional 



242 



KINEMATICS OF MACHINES. 



to that through which the steering-wheel has been turned; 
the motion of the steering-engine must then cease and the 
rudder must be held until the steering-wheel is again moved. 




Fig. 162. 
Many kinds of lifting and hoisting mechanisms form varia- 
ble escapements of this type ; another well-known example 
is the hydraulically controlled steam reversing- gear, often 
applied to large marine engines.* 

* See also Fig. 185, § 89. 



RATCHET MECHANISMS AND ESCAPEMENTS. 243 

As an example of an adjustable escapement of a more 
simple kind, the "Weston Triplex" pulley-block has been 
selected.* It is shown diagrammatically in section in Fig. 
162. The link a forms the body of the block and has on it the 
bearings of the rotating chain-wheel 6, with which the hoist- 
ing-chain engages. By means of an epicyclic train hace (in 
which the fixed annular wheel forms part of the link a), ^ is 
driven by the rotation of a central shaft e. The hand-chain 
drives the wheel /, which works on a fine-threaded screw cut 
on e, in such a way that, when screwed up, a flange g^ is 
compressed between the face of / and a corresponding flange 
e^ secured rigidly to e. A friction-clutch is thus formed. 
The flange g^ forms part of a ratchet-wheel g, connected with 
a by a roller ratchet like that of Fig. 152. The action of the 
block may be summarized thus : 

(i) Hoisting. The hand-chain wheel / screws up on g^ 
and turns the shaft e by means of the friction-clutch fg^e^^. 
The ratchet gear gka runs freely. 

(2) Standing. On ceasing to hoist, the load on the 
hoisting-chain tends to turn e in the reverse direction; the 
ratchet gear engages and holds the load. 

(3) Lowering. On turning / in the reverse sense, g^ 
being held by the ratchet, / is screwed back on its thread, 
the friction-clutch is released, and the load is lowered so 
long as f is kept in motion. On stopping / the m.otion of e 
at once screws up the clutch and checks the load. 

The contrivance is thus seen to consist essentially of a 
rotating shaft driven by an automatic friction-clutch and 
held by a stationary friction-ratchet, the whole forming an 
adjustable frictional escapement started and released at 
will, the motion of the central shaft imitating that of the hand- 
chain wheel. It should be noticed that the action of the 
machine differs in this important respect from that of a sim- 
ple hoisting-block provided with an ordinary friction-brake. 

* See Engineering, August 22, 1890. 



CHAPTER X. 
MECHANISMS INVOLVING NON-RIGID LINKS. 

79. Non-rigid Links. — In giving a definition of a machine 
or of a mechanism we were careful to use the word ' * resist- 
ant ' ' as apphed to the material forming the links composing 
the mechanism. Many essential portions of actual ma- 
chines are non-rigid, but are nevertheless resistant, and 
their occurrence, while it does little to complicate the 
machine from a kinematic point of view, often introduces 
dynamical problems of the greatest interest and difficulty. 
The different classes of non-rigid links, and pairs involving 
them, have already been noticed; w^e have now to study 
certain kinematic questions arising from their use. 

In considering non-rigid links in mechanisms or machines 
it is necessary to take account of the way in which their 
form changes while in motion. One class of these Imks 
is composed of those which, while very yielding as far as 
bending or thrusting actions are concerned, do not change 
their length appreciably when a direct pull is applied. 
Belts, ropes, and chains, which come under this head, are 
therefore often of great use in machines where energy has 
to be transmitted in changing directions. This is usually 
done by causing the flexible tension-links, in the form of 
belts, ropes, or chains, to pair with, and communicate 
motion to, rotating drums or wheels. On account of their 
change of form, non-rigid links can have no virtual axes or 
virtual centres. 

80. Velocity Ratio in Belt-gearing. Length of Belts. — 

The linear velocity of a rope or belt passing over two or 

244 



MECHANISMS INFOLVrNG NON-RIGID LINKS. 



245 



more pulleys may be considered for kinematic purposes as 
being the same throughout its length. In practice the 
stretching of a rope or belt under load often has an appre- 
ciable effect on the velocity ratio of the pulley it drives; 
we shall here treat questions of velocity ratio as if the 
belt or rope were inextensible. Fig. 163 represents a pair 
of cylindrical pulleys connected by a belt, which may be 
"open" or /'crossed" so that the pulleys rotate either in 
the same or in opposite senses. We shall for the present 
neglect the effect of the thickness of the belt or rope. 




Fig. 163. 



In these cases if V be the linear velocity of the belt and 
R^Rb "t^^ radii of the pulleys, the angular velocity ratio will 
evidently be found from the relation 

R'V R' 



CO. 



OJr 



246 KINEMATICS OF MACHINES. 

the negative sign corresponding to the case of a crossed belt. 
It is, of course, assumed that there is no sHpping. 

The length of a belt is easily expressed in terms of the 
radii and the distance d between the centres of the pulleys. 
The total length of belt not in contact with the pulleys is 

2^/d'-{R^±R,y, 

the negative sign here corresponding to the case of an open 
belt. If d be the angle that the straight part of the belt 
makes with the centre line of the pulleys, then the length of 
belt in contact with the pulleys will be 

{7t-\-2d){R^-{-R^ for a crossed belt 

and {7i + 2d)R^-\-{7t-2d)R^, 

or 7:{R^^R^) + 2d(R^-R^) for an open belt, 

where d. = sm" ' °^ \ 

The expression for the total length of belt will then be 
for an open belt 

2Vd'-{R,-R,r + n{R, + R,) + 2(R,-R,) sin-'^^\ 
and for a crossed belt 

2Vd^-{R^ + R,r+{R^-^R,)(7r + 2 sin-^--^^y 

It will be seen that the length of a crossed belt is thus 
constant so long as the sum of the radii and the distance be- 
tween the centres of pulleys are constant quantities. 

81. Belt-gearing for Variable Velocity Ratio. — Fig. 164 
shows the arrangement of ''cone pulleys" employed in 
driving machinery so as to render it possible, by shifting a 



MECHANISMS INVOLVING NON-RIGID LINKS. 



247 



belt from one pair of steps to another, to obtain at will any 
one of several velocity ratios. It is plain that the same 
crossed belt will run with the same tightness on any pair of 
steps so long as the sum of the radii of each pair is the same. 
An open belt, however, is generally required, in which case 
the tension will be different on each pair of steps, unless 




Fig. 164. 

their diameters are specially calculated. Approximate 
methods for readily doing this have been devised,* while 
Reuleaux f gives a rigorous graphical treatment of the prob- 
lem. Referring to Fig. 163, we have as an expression for 
the length of an open belt 



/ 



dcosd+^(R^-^R,) + d{R^-R,)]. 



*Umvin, Machine Design, Vol. I, p. 373 ; Smith, Trans. Am. Soc. M. E., 
Vol. X, p. 269. 

f Reuleaux, The Constructor. Trans, by Suplee, p. 189. 



248 KINEMATICS OF MACHINES. 

Now R^ — R^^ = d sin 6 \ therefore 

, l--=2\^dcosd + ~i2R^-dsmd)-{-dds\nd\ 



and 



R= (cos/9 + 6'sin(9) + . 

"•27:71 ^2 



Similarlv, 



i^. = — 



/ d 



27t 7t 



(cos (9 + ^ sin d) 



d sin ^ 



Take a pair of rectangular axes OA and 05, (Fig. 165) 
and make OA =d. Draw a curve CD, the involute of the 
circular arc AC, having as its centre. Then, since the 




10 INCHES 



Fig. 165. 
angle 6 must lie between 0° and 90°, it must have some such 
value as COE, in which case the line EF, tangent to y4C 
at Ey and cutting the involute at F, has a length equal to 
the arc EC. Hence EF ^ 6d, and, drawing KFH parallel 
to AO, we have GH = EF sin d, and 

OH=OG+GH 

= d(cosd-\-dsmd). 

Next make 0B = 7id and join AB. Draw ^D parallel to 
OB. Let HE meet AB in V and AD in K ; then 

OH OB 7z' 



MECHANISMS INVOLVING NON-RIGID LINKS. 249 

d 
and therefore KV = - (cos d-\-d sinO). 

Again, if we set off AL=-, and draw LM parallel to AO. 

2 

and cutting AB in M , we have 

27Z 

Draw MN parallel to BO and cutting HK in A/", then 

VN=KN-KV 

Id 
= (cos 6 + sm 6). 

271 7Z 

_ , . , , ^ d sin 6 -, 1 -, 

io obtain the value of we need only draw a semi- 

d 
circle OQP having a diameter - ; then 

Finally a curve DRST may be drawn by setting off VR = 

VS = 0Q, and repeating the construction as required. This 

gives 

NR==VN+VR 

I d ^ ^ ^ . ^s dsind 
= -(cos^ + ^sin^)+ 

27r TT 2 

and NS = VN-VS^ 

I d , ^ ^ . ^s d sin 6 
= -(cos ^4- ^ sm ^) 

2Tl 7t 2 

Thus R^-R, = VR+VS=SR. 

Plainly for given values of / and d we can determine R^ 
and Rj^ for any value of 6 (or for any required velocity ratio) 
by the aid of the curve DRST. 

In practice it is usual to find that the diameters of the 



250 



KINEMATICS OF MACHINES. 



first pair of steps or their radii, R^ and i?^, are given, together 
with d, the distance between centres of pulleys. The prob- 
lem then is to find the radii of another pair of pulleys, RJ 
and R^\ on which an open belt of the same length will run 

7? ' 
with a given velocity ratio -^,. The author has found the 

following a convenient method of utilizing the Reuleaux 
diagram for solving this problem, and for finding inci- 
dentally the length of belt required. This length, however, 
is not often necessary, as it is more easily measured from 
the pulleys when finished and in position. 

Draw the rectangle AOB and the curve DRST exactly 
as described above, and as shown in Fig. 165, making OA, 
say, 10 inches in length. This diagram can be used for find- 
ing pairs of radii of steps having any desired velocity ratio, 
and the lengths of these radii will be obtained in terms 
of d, the distance between the shaft centres. Having ex- 
pressed R^ and jR^,, the given pair of radii, in terms of d, 
it is easy, by applying a scale of inches and hundredths to the 
diagram, to determine that position of the line SR which 
will give the proper value to R^ — R^. The length RN is 
then measured to the proper scale and the point N found. 
If required, the half length of the belt is then settled by 
drawing the line 7VM, and the next thing is to find another 
set of points R\ S\ N' such that R'N' and 5W will have the 

ratio required for the radii of 
the next pair of steps. This is 
readily done by drawing on 
tracing-paper a set of radiat- 
ing lines (Fig. 165a), VX, VY\ 
VZ, arranged so as to cut all 
lines perpendicular to VZ in the 

R\ 




required ratio, namely. 



R' 



It 



Fig. i6sa. 



is convenient to draw another 



line, y y, such that lines perpendicular to VZ are also cut in 



MECHANISMS INyOLyiNG NON-RIGID LINKS. 251 

the ratio -^. By applying this diagram to Fig. 165, the three 

points R\ S\ A/"' can readily be pricked off in their proper 

positions. When measured to the proper scale, i?W and 

S'N^ give the values of the pair of radii required. In Fig. 

R ' 

165 the ratio ^, is 12.0, while 

J<b 

^ is 6.0 and l = K.66d. 
^b 
If the real value of d is taken as 30 inches, while R^ and i?^ 

are 25.2 and 4.2 inches respectively, the diagram gives for 

RJ and -R/ the values 26.4 and 2.2 inches. An open belt 

of about 170 inches in length would run on either of these 

two pairs of pulleys. 

It should be noted that when d is large in comparison 
with the size of the step pulleys, it is often sufficiently ac- 
curate to proportion the latter as if intended to run with a 
crossed belt ; for this purpose the sum of the radii may be 
made constant. 

To make allowance for the effect of the thickness of the 
belt or rope in our calculations it is only necessary to reflect 
that we have really taken the thickness of belt as being 
negligible when compared to the diameter of the pulley. 
In practice this is frequently not the case. Suppose, for 
example, that a belt whose thickness is a quarter of an 
inch is running on a pulley 6 inches in diameter. We 
assume that while passing round the pulley the layer of 
material at the centre of the thickness of the belt is neither 
stretched nor shortened, so that the arrangement will be 
equivalent kinematically to a pulley 6j inches in diameter 
on which a belt of negligible thickness is running. In other 
words, we take it that the effective radius of the pulley is 
in all cases to be measured to the centre of the thickness of 
the belt or rope. 

82. Velocity Ratio in Chain- and Rope-gearing. — Rope- 
and chain-gearing is extensively used for the transmis- 



252 



KINEMATICS OF MACHINES. 



sion of power, as well as in machinery for hoisting, wind- 
ing, and lowering. In many cases it is necessary to 
provide the rope-drums or pulleys with guiding or re- 
taining grooves. The various forms 
of rope and chain tackle are too 
familiar to require extended notice 
here; the ratio of the speed of the 
rope to the speed of the body 
moved by the tackle can always be 
readily found. As an example, we 
may take the Differential Pulley- 
block of Fig. 1 66. In this case the 
upper block has two sheaves a and a' 
rigidly connected or made in one 
piece; the chain is prevented from 
slipping on these sheaves by suitable 
projections in their grooves. Evi- 
dently on hauling in the sense shown 
by the arrow, the loop or bight of the 
chain passing around h will be short- 
ened during each revolution of a and a' by an amount 
equal to the difference of the circumferences of those 
pulleys. Hence, if we call R^ and R^ the effective radii of 
a and a' we shall have 

speed of chain _ 27tR^ 

speed of hoisting J ( 2 nR^ — 2 nR) 

2R^ 

R1-R2 
Sometimes it is desirable to arrange hoisting gear in such 
a way that the velocity ratio is variable. For instance, in 
the winding gear of a deep mine it is necessary to wind the 
rope on a drum of continually increasing radius provided 
with a spiral groove, so that when one cage is at the bottom, 
its weight together with that of the attached rope may be 
balanced by the smaller weight of the other cage alone 
acting on a portion of the drum which is of larger radius. 




Fig. 166. 



MECHANISMS IN^OL^ING NON-RIGID LINKS. 



253 



A similar device is employed in the "fusee "of a chro- 
nometer. 

In some cases shafts and pulleys are so connected by 
chain-gear that their velocity ratio is not uniform through- 
out the revolution. Fig. 167 shows one form of sprocket- 
wheel and chain. The wheel is 
furnished with teeth engaging 
with the links of the chain and 
effectually preventing slipping; 
these teeth should evidently 
have profiles composed of cir- 
cular arcs parallel to the paths 
described by the centres of the 
pins as they move relatively to 
the wheel. On considering a 
pair of such wheels connected 
^' by a chain it will be seen that 

if their pitch-circles are of unequal diameters, their velocity 





Fig. 167a. 
ratio will not be the same in every position. Fig. 167a 
represents the centre line of a chain connecting a pair of 
sprocket-w^heels ; the wheels have four and eight teeth re- 
spectively. When in the position shown by full lines, the 
pair of wheels and the chain are equivalent to a four-bar 
mechanism or quadric crank chain A BCD. Applying the 



254 KINEMATICS OF MACHINES. 

construction of § 28 we find that the velocity ratio is 

co^ BC 



CO, AD' 



But when in the position shown by dotted lines the velocity 
ratio is 

w^^FB BC 

w,~AE^AD' 
the equivalent position of the quadric crank chain being 
AEGB. In sprocket- and chain- gearing as used for cycles 
this inequality of velocity ratio may amount to from 5 to 1 5 
per cent. 

A form of chain used by Mr. Hans Renold for transmitting 
power between two parallel shafts is shown in Fig. 168. The 




Fig. '168. 

chain links have projections or teeth on their inner edges, 
so formed as to gear with teeth on the wheel rims. It will 
be seen from the diagram that these teeth profiles, the work- 
ing portions of which are made up of straight lines, are so 
arranged that the links enter and clear the wheel teeth with- 
out appreciable sliding or rubbing motion. The angle em- 
braced by the sides of the wheel-tooth profile is smaller the 
smaller the number of teeth in the wheel. Chains of this kind 
will work correctly even if slight stretching has taken place. 
The periodical inequality of velocity ratio when the driving 
and driven wheels are of different sizes is to be determmed 
for these chains in exactly the same way as for ordinary 
pitch chains, the effective diameters of the wheels being, of 
course, measured to the centres of the pins of the chain links. 
83. Belt- and Rope-gearing between Non-parallel Axes. — 
It should be noted that belts and ropes may be used for 



MECHANISMS INVOLVING NON-RIGID LINKS. 



255 



transmitting power between shafts whose axes are not par- 
allel ; in some cases idle guide-pulleys- are required in order 
that the belt or rope may run satisfactorily. For this to be 
the case one condition must be fulfilled, namely, that where- 
ever a belt or rope is running on to a pulley the centre line 
of the advancing belt or rope must lie in the central plane 
of the pulley on to which it is running; i.e., in a plane nor- 
mal to the axis and passing through the centre of the pulley.* 
A number of cases of belt transmission between non-parallel 
axes are illustrated here. The belt in Fig. 169a can only 




Fig. 1690!. 
be run in the direction indicated by the arrows, the portion 
RT lying in the plane of the pulley B, but not in that of A, 
while the part SU lies in the plane of A. Similar remarks 
apply to Fig. 1696, where, however, a guide-pulley is used. 
In Fig. 169c, it will be seen that if the guide-pulley C is 
placed in a plane containing the parts of the belt 5 T and UT 
every straight portion of the belt lies in a line which forms 

*See Webb, Trans. Am. Soc. M. E., 1883, p. 165. 



256 



KINEMATICS OF MACHINES. 



the intersection of the planes of its pulleys; hence the belt 
will run either way. Fig. i6gd shows the general case when 
the axes are inclined. Any two points, X and Y, are chosen 
on the line forming the intersection of the planes of the 
pulleys A and B, and tangents XR, XS, VT, YU are drawn 
to these pulleys. The guide-pulleys C^ and C^ are then 




Fig. 1693. 
placed in the planes Xi?5, YTU' respectively. Under these 
conditions the belt will run either way. These examples will 
serve to indicate the method to be adopted in arranging 
belt-gearing when the axes of the shafts are not parallel. 

Similar remarks apply to arrangements for rope-gearing, 
but in this case, as the pulleys are grooved, guide-pulleys are 
not so frequently required. 

84. Springs. — While belts, ropes, and chains are espe- 
cially of use for transmitting energy, the flexible links known 
as springs are of service where energy has to be stored up 
and again restored when required. Problems connected 



MECHANISMS INyOLyiNG NON-RIGID LINKS. 



257 



with mechanisms involving springs will in general deal with 
questions of Dynamics rather than with questions of Kine- 
matics; it will be sufficient here to notice some cases in 
which the energy of springs is employed for kinematic pur- 
poses; i.e., for controlling or assisting the relative motions 
of machine parts. 




Fig. 169^. 

Springs are often used for the closure of mechanisms and 
pairs. The spring h, for example, in Fig. 161 supplies the 
force required to keep b in contact with e or in contact with 
c, as the case may be. Certain springs in gun-locks fulfil a 
similar purpose, and springs of the same kind form an essen- 
tial feature in most ratchet mechanisms and escapements. 



258 



KINEMATICS OF MACHINES. 



In Fig. 170 the essential parts of the lock of a Winchester 
rifle are shown. The lock mechanism contains two springs ; 
the main-spring a is bent when the hammer h is drawn back 




Fig. 169a'. 
and cocked, and the • energy stored in this spring is availa- 
ble, when released, for striking the firing- pin c and thus 
exploding the cartridge. The trigger-spring d bears on the 
trigger e, which serves in fact as a pawl or detent for the 
hammer. When the hammer is at half-cock the point 
or nose of the trigger enters the first notch on the 
hammer ; the hammer is then secure, as the form of the 
notch prevents the trigger from being pulled. When 



MECHANISMS INVOLVING NON-RIGID LINKS. 259 

the hammer is placed at full cock, however, the point 
of the trigger engages with the second notch, which 
is of such a form that the trigger can be pulled and the 
hammer released. While both these springs may be re- 
garded as serving for purposes of closure, d has no other use ; 
a, on the other hand, stores up energy in the way already 
described. The whole mechanism forms a locking and re- 
leasing ratchet-train (see§ 77) which is spring-closed. 

In many cases springs are used simply as means of storing 
energy, a very familiar example being the coiled spring 




Fig. 170. 

which drives a clock or watch; in other instances they are 
employed simply to control or modify the relative motion 
of machine parts. The springs in a buffer, or the springs 
which hold down a safety-valve, come under this heading. 

85. Fluid Links. Pressure Pairs. — It has already been 
noted that we class under the name of spring those portions 
of mechanisms whose elastic deformations, when the mech- 
anism is in action, are considerable as compared with the 
dimensions of the spring itself, and with the extent of motion 
of the other links of the mechanism. According to this 
definition we ought to include in our list not only solid 
springs, but also such bodies as the air in an air-compressor, 
which, although fluid, suffers elastic deformation. 

Hydraulic machines, again, contain fluid links which do 



26o 



KINEMATICS OF MACHINES. 



not sensibly change their volume under the pressures to 
which they are subjected in working. Hence fluid links may 
be divided into (a) elastic and (6) non-elastic links. As 
has been previously stated, the changes of form and volume 
of these links involve questions of dynamics which lie out- 
side of the scope of the present work, so that we shall here 
consider only in their kinematic aspect certain mechanisms 
containing fluid links. 

In every case the pairing of the fluid link with the solid 
link or links containing it will be ** pressure pairing" (see 
§ 9) ; in using fluid links in mechanisms we therefore meet 
with a constructive difQculty not found when employing 
rigid material only, namely, that all moving parts in contact 
with fluid under pressure have to be so made that no unneces- 
sary leakage can take place. The means of attaining this 
object is not important from a purely kinematic point of 
view. 

A large number of mechanisms containing fluid links will 
be found to have as their counterparts mechanisms con- 
taining rigid links only. We find, for example, many in- 




FiG. 171. 

stances of fluid ratchet -trains, and Fig. 171 compares the 
well-known hydraulic press with a system of rigid levers 
whose ends AB have the same velocity ratio as the plunger 
C and ram D of the press. By suitably proportioning the 
areas of C and D we can obtain any desired ratio of the 
rising velocity of D to the falling speed of C, for it is plain 



MECHANISMS INVOLVING NON-RIGID LINKS. 261 

that the volume of water displaced in a given time by C 
must be equal to that entering the cylinder in which D 
works. Hence the speed of D will be to the speed of C 
inversely as their areas, assuming that the fluid is incom- 
pressible and that no leakage occurs. 

It is evident that under these circumstances the relative 
velocity of C and D will be unaffected by the length or form 
of the pipe or passage communicating between the cylinders. 
These might in fact be separated by a considerable distance, 
in which case the mechanism would serve for the hydraulic 
transmission of energy. Such transmission is found of 
great utility under certain conditions.. Similarly arrange- 
ments for the transmission of power by- compressed air have 
been devised ; in either case it is the fluid link which renders 
this type of transmission possible and economical. 

86. Chamber Crank-trains. — The most important appli- 
cation of the crank-chain in machine construction is its use — • 
in kineniatic combination with a fluid link — for- the purpose 
of an '^engine" or prime mover, or for the purpose of a 
*'pump" or machine for moving or compressing the fluid. 
The fluid link may consist of stearn, air, gas, or water, and 
the mechanism must include a suitable chamber for enclos- 
ing it. We proceed to give a few examples of such chamber 
crank-trains, selected from the numberless instances of every- 
day occurrence. 

Any of the inversions of the slider-crank chain of Chap- 
ter IV may be converted into a chamber crank-train if we 
make one of its links into a vessel or chamber and convert 
another link (in some cases two others) into a plate or dia- 
phragm moving in the vessel in such a fashion that the fluid 
link occupies the space enclosed. As the mechanism oper- 
ates, the effective volume available for the fluid link is 
changed and the fluid expands, or is compressed; it enters 
or leaves the vessel or chamber in conformity with the alter- 
ation in volume and the conditions under which the ma- 
chine is working. The ordinary direct-acting steam-engine 



262 



KINEMATICS OF MACHINES. 



is a familiar example and is derived from the turning slider- 
crank of Fig. 60. The link c forms the piston or movable 
diaphragm, while the link d takes the shape of the cylinder 
in which the piston travels. In some instances the cylinder 
or chamber is so formed as to enclose partially or completely 
the links a and h. Fig. 172 shows diagrammatically how 
this is done in the case of the small enclosed petroleum or 
gasoline motors so much used for the propulsion of auto- 
mobiles and boats. In Fig. 68, again, the swinging block 
slider-crank is used as an oscillating steam-engine ; c is now 
the cylinder and d the piston and rod. 





Fig. 172. 



Fig. 173. 



It is also possible in the swinging block slider-crank to 
convert the fixed link into the chamber. Fig. 173 shows 
a method of doing this, suggested by Reuleaux. On con- 
sideration it will be obvious that this arrangement, like 
many others quite possible kinematically, will not be likely 
to give satisfactory results in practice. The mechanical 
difficulties of making the contact between c and h sufficiently 
good to avoid leakage, and at the same time so free as to 
avoid frictional loss, are so great that a large number of 
possible chamber crank-trains are of no practical value. 



MECHANISMS INVOLyiNG NON-RIGID LINKS. 



263 



Passing on to the turning block slider-crank chain of Fig. 
73, this has been converted into a chamber train, and was 
originally proposed as a steam-engine by Lord Cochrane 
in 1 83 1 and 1834. Probably the inventors of so-called 
** rotary" engines and pumps have nowhere found so exten- 
sive a field for their ingenuity as among mechanisms derived 
from this kinematic chain. One form of the Cochrane en- 
gine is shown in Fig. 174. Here the rotating chamber is 
formed from the link h of Fig. 73a, and has line contact 
with the link d. The fixed link a (corresponding to the 
crank in a direct-acting engine) forms the frame or support 
of the mechanism, and the working fluid expands in the 
spaces enclosed between 6, c, and d. 

The swinging slider-crank when used as a chamber train 
has already been shown in Fig. 74. 




Fig 174. Fig. 175. 

From the double slider -crank chain a number of chamber 
trains can be derived. Fig. 79 shows one form — the com- 
mon donkey-pump. In Fig. 175 we have a form of steam- 
engine proposed by Root in 1864 and since revived by other 
inventors. On comparison with Fig. 77 or 79 the corre- 



264 KINEMATICS OF MACHINES, 

Spending links will be readily recognized. In the Root 
engine the rectangular spaces enclosed between the links 
c and d and between h and c are used as chambers to receive 
the working fluid; their varying volume serves for the ex- 
pansion of the steam. It should be noted that this form 
of engine does not involve the use of higher pairing, and 
there is not the difficulty experienced in so many chamber 
crank-trains in preserving a steam-tight joint. When the 
surfaces between a, b, c, and d are adjusted so closely as to 
avoid leakage, there is in practice found to exist a consider- 
able amount of friction, and the accuracy of adjustment is 
easily destroyed by expansion due to any slight local varia- 
tions in temperature of the different parts of the engine. 
It is for reasons of this kind that no form of chamber crank- 
train has yet been able to compete in practice with those 
types derived from the turning slider-crank. 

It is, of course, to be understood that in a chamber crank- 
train used as a motor or pump suitable provision must be 
made, for the government of the admission and outflow of 
the working fluid. This is sometimes effected by arranging 
the necessary openings and passages so that they are opened 
or closed by the motion of the solid links themselves. More 
frequently it is necessary to provide a subsidiary ratchet- 
train or valve-gear, which forms no essential portion of the 
original machine, if we consider only the motion and pairing 
of the solid links, the object of the valve-gear being simply 
the control of the fluid link. These mechanisms are con- 
sidered further in § 89. 

87. Chamber Wheel-trains. — Reuleaux divides motor- 
mechanisms containing ' ' pressure organs " or fluid links into 
two classes. We have first those mechanisms in which the 
motion is more or less intermittent, so that the whole ma- 
chine forms a ''fluid ratchet-train." The kinematic chains 
discussed in the last section, when provided with the neces- 
sary valve-gear, belong to this class. The second class 
includes those ' ' running mechanisms ' ' in which the motion 



MECHANISMS INVOLVING NON-RIGID LINKS. 



265 



of all the solid links is continuous, and we now proceed to 
consider some examples of this kind of chain, formed by 
modifying certain wheel-trains in such a way as to constitute 
a chamber wheel-train. The chamber is, in general, formed 
from the frame of the wheel-train and carries the wheels by 
means of simple turning pairs. The fluid or working sub- 
stance occupies the space between the wheels and the cham- 
ber, and such mechanisms, in spite of certain mechanical 
disadvantages, are often used as pumps or motors, or as 
meters for measuring the amount of fluid passing through the 
chamber. 

The chamber wheel-trains which are simplest from a 
kinematic point of view are those containing only one wheel 
and the necessary chamber and passages for the guidance of 
the fluid link. In Fig. 176, for example, is shown diagram- 




FiG. 176. 

matically a centrifugal pump, the whole mechanism consist- 
ing of the pump-casing a, the wheel and shaft 6, and the 
fluid c. A turbine, or water-wheel, of course falls into the 
same class. 

Figs., 177a, 1776 and 177c show three types of chamber 
wheel-gear amongst many which have found some degree of 
favor in practical use as pumps or motors. Fig. 177a is the 



266 



KINEMATICS OF MACHINES. 




Fig. 177a. 




Fig. i77<^. 



MECHANISMS INyOLyiNG NON-RIGID LINKS. 



267 



Pappenheim chamber wheel-train, consisting of a pair of 
equal toothed wheels, having continuous tooth-contact, while 
the points and sides of the teeth fit as closely as possible to 
the walls of the chamber. This train has been utilized to a 
considerable extent as a pump for water, the volume dis- 
charged per revolution being evidently equal (if leakage 
is neglected) to twice the volume of the tooth-spaces of one 
wheel. Such a pump is, of course, most suitable for running 
at a high speed and against a low pressure. Fig. 177& 
represents the well-known Root blower, the action being 
identical with that of the mechanism of Fig. 177a. The 
wheels in this machine have, however, only two teeth each, 
and external gearing is required to maintain constant contact 
between the teeth. Epicyclic chamber wheel-trains are some- 
times employed. Fig. 1 77c represents the mechanism of the 




Fig. 177^. 



Hersey water-meter. Here one wheel forms the case a, while 
the rotating shaft b (not shown) carries an eccentric -pin on 
which works the ' ' rotary piston ' ' or wheel <:. It will be at 
once seen that this arrangement is really an epicyclic train. 

Inventors have eagerly sought to discover some kind of 
chamber wheel-train in which the more or less imaginary 
disadvantages of the reciprocating engine or pump are 



268 



KINEMATICS OF MACHINES. 



-d 



d^ 





avoided. Almost every conceivable form of such gear * has 
been invented and reinvented and used with varying suc- 
cess. No enthusiast, however, has yet succeeded in pro- 
ducing a machine which is a serious competitor with the 
ordinar}^ direct-acting engine or pump formed from the 
slider-crank chain when used for the same kind of work. 

88. Ratchet- trains Containing Non-rigid Links. — The 
classification of Ratchet Mechanisms in general has been 
considered in the last chapter ; we have now to study exam- 
ples of those ratchet-trains which contain non-rigid links. 

Fig. 178 shows diagrammatically 
the construction of a common lift- 
pump. On comparing Figs. 178 
and 153 we find in each case a 
body b to be raised or lifted by the 
action of a pawl c, moved by a 
running-ratchet train. In the 
pump this pawl takes the form of 
a non-return valve carried in the 
pump-bucket d. The body of the 
pump corresponds to the frame a 
in Fig. 153. Further, it is plain 
that to keep the fluid in the pump 
from running back we must provide 
a foot-valve c^ corresponding ex- 
actly to the pawl c' in Fig. 153. 
We have here in fact a checking-ratchet train abc^ exactly 
as in the mechanism of Fig. 153. The common lift-pump is 
then a combination of two ratchet- trains acting on the same 
Hnk, and this link is the fluid which is being pumped. 

In pumps special devices are often necessary to obtain 
a more continuous motion of the fluid than is possible with a 
single-acting ratchet-train. Fig. 179 shows diagrammatically 
one example of this — a pump of a type sometimes used 



a 



a 



Fig 178. 



* See Reuleaux, Kinematics of Machinery, Chapter X; Burmester, Lehrbuch 
der Kinematik, §§ 96-109. 



MECHANISMS INVOLVING NON-RIGID LINKS. 269 

for operating a hydraulic accumulator. Here the pump- 
piston d is provided with an enlarged rod d^ of cross-section 
approximately equal to one half the area of the piston or 
bucket. Thus during the stroke from left to right one half of 
the fluid passing the valve c^ is compelled to issue through 
the valve c^, and one half enters the pump-barrel. During 
the reverse stroke this remaining half is expelled, and 
another volume of fluid enters the pump through the valve 
Cj. This ''differential" pump, therefore, gives a fairly con- 
tinuous discharge, but differs from a double-acting pump 




Fig. 179. 

in that its suction is not continuous, but only occurs during 
one stroke of the bucket. We have in this case an example 
of the combination of three running ratchet- trains. 

In Fig. 180 is shown an ingenious form of pump in 
which only one set of valves is required, the pump-piston 
itself performing the function of a releasing ratchet. The 
Edwards air-ptmip is used for pimiping air and water from 
the condenser of a steam-engine. The bucket or plunger 
P has no passage through it, and during the downward 
stroke, while the head valves V are closed, the pressure in the 
space A is reduced, so that air passes in from the condenser 
through the ports BB as soon as these are uncovered. At 
the same instant the plunger on reaching the bottom of its 
stroke has displaced the water from the bottom of the pump 
and has driven it through the passage C into the space above 



270 



KINEMATICS OF MACHINES, 



the bucket. The air and water are then discharged on the 
upward stroke. It will be seen that we have here a releas- 




FiG. 180. 



ing-ratchet train in which the ratchet (the pump-plunger) 
itself propels a portion of the fluid to be moved, and also 
prevents it from returning. When at the lowest point of its 
stroke the piston in uncovering the ports has acted as a 




Fig. 181. 



driving and releasing ratchet, first opening the passages B 
and then propelling the fluid through them. 

From this point of view we may regard any sliding-valve 



MECHANISMS INVOLyiNG NON-RIGID LINKS. 271 

as a checking and releasing ratchet. Fig. 181 shows a lon- 
gitudinal section through the cylinder of a steam-engine 
provided with a piston-valve, and it will be seen that this 
valve uncovers the steam-ports and admits and cuts off the 
steam just in the same way as the bucket of the Edwards 
pump uncovers its ports. The slide-valve of a steam-engine 
is, however, only a checking and releasing ratchet ; it has no 
part in propelling the fluid. 

Valves and cocks are frequently employed as brakes, 
and they then form parts of f rictional ratchet- trains in which 
the moving link is a fluid. In Fig. 182 we have a diagram 

c 




Fig. 182. 



of an arrangement used to control the longitudinal move- 
ments of a rod R. A piston P works in a closed cylinder 
filled with fluid, and the two ends communicate by means 
of a passage which can be wholly or partially closed by 
rotating a cock C. A valve would, of course, answer the 
purpose. Here the partial closing of the cock or valve 
opposes a f rictional resistance to the movement of the fluid, 
and therefore also to the movement of the piston, and in fact 
the cock or valve acts as a checking ratchet. Somewhat 
similar arrangements may be devised to act as hydraulic 
brakes in the case of rotating shafts ; the well-known Froude 
brake is an example where the necessary resistance to the 
motion of an engine-shaft is obtained by attaching to the 
shaft a special rotary pump which discharges its water 
through a small passage. 

Ratchet-trains often contain belts or other flexible links. 



272 



KINEMATICS OF MACHINES. 



The strap brake of Fig. 183 may be looked upon as a fric- 
tional checking-ratchet train. The fixed link of the train is 
not shown, but it will be easily seen that the strap corre- 
sponds in function to the brake-block and lever of Fig. 159. 
Flexible links are occasionally used in clutches, which, 




Fig. 183. 




Fig. 184. 

as we have already seen (§ 77), are really stationary ratchet 
trains. Fig. 184 shows the longitudinal section of a coil 
clutch, whose purpose is similar to that of the friction clutch 

of Fig. 159a. 

The action of the contrivance may be explained as follows : 
The driving shaft A has firmly secured to it the hollow 
drum Aj, inside which is a metallic coil A^ loosely enclosing 
the pulley B^, which is keyed to the driven shaft B. One 
end of the coil (at C) is fixed to A„ the other end has upon it 



MECHANISMS INVOLVING NON-RIGID LINKS. 273 

a radial projection and can be rotated so as to cause the 
coil to grip the outer surface of B^. This rotation is ac- 
complished by slightly turning the lever D on its pin by the 
aid of the sliding sleeve E, which can be moved along 
the shaft by means of a fork engaging in its groove. When 
pushed in, the conical end of E pushes aside the lower arm of 
the lever D and closes the coil. Such a clutch will only 
drive one way, but the numerous turns of the coil on the 
drum give it enormous frictional resistance, and the end 
pressure on the shaft A is not large. The whole arrange- 
ment forms a frictional ratchet-train. 

89. Pressure Escapements Containing Fluid Links. — We 
have classed under the term escapements certain checking- 
and releasing-ratchet trains which are so arranged that the 
moving link is alternately released and checked by the 
action of the mechanism itself. Escapements containing 
fluid links form a class of machines which are of the utmost 
importance industrially, and some examples of such mech- 
anisms will now be considered, following the nomenclature 
of § 78. It was there shown that escapements are really 
ratchet-trains which work automaticall}^, and in the same 
way a self-acting fluid-ratchet-train may be said to be a 
fluid or pressure escapement, the driven or moving link 
being the working fluid. 

A steam-engine or fluid motor which is provided with a 
governor regulating and rendering uniform its rate of motion 
obviously answers to our definition of a uniform escapement. 
In a properly governed steam-engine or gas-engine we may 
compare the function of the governor with that of the pen- 
dulum or balance-wheel of a clock or chronometer, while 
the escapement is evidently represented by the valve-gear. 
The valves themselves control the range of movement of 
the working fluid, exactly as the ratchets in a clock escape- 
ment control the range of movement of the escape- wheel. 

Periodical fluid escapements are not of frequent occur- 
rence. We may perhaps class under this head such contriv- 



274 



KINEMATICS OF MACHINES. 



ances as gas- and water-meters, which vary their rate of 
motion in proportion to the quantity of fluid passing per unit 
of time. 

Adjustable or variable fluid escapements are of consider- 
able importance. A large number of pressure mechanisms 
corresponding somewhat in their mode of action to the 
hoisting machine described in § 78 are used as steering- or re- 
versing-gears. Such a contrivance consists essentially of a 
controlled motor (moteur asservi) so arranged that when 
started by the admission of the working fluid the motor 
itself closes the admission-valve, and therefore stops unless 
the controlling valve is still further opened by hand. This 




Fig. 185. 

will perhaps be made clear by an example. The slide-valve 
controlling the admission and exhaust of steam to the cylin- 
der in Fig. 185 is connected to a system of levers as shown. 
When the lever 6' is moved by hand in the sense show^n by 
the arrow, steam is admitted through the port P^ while the 
port P^ is placed in communication with the exhaust. The 
piston c moves in response, and, if the lever c' is properly 
proportioned, gives the valve b a backward motion exactly 
equal to the forward movement it received from the hand 



MECHANISMS INVOLVING NON-RIGID LINKS. 275 

lever 6'. Thus the piston c follows the motion of the hand- 
lever h\ It will be obvious that kinematically this mechan- 
ism is of the same general class as the hoist previously 
described, and is accordingly an adjustable escapement.* 

A large number of fluid-ratchet-trains and escapements 
are discussed by Reuleaiix.t 

* For a description of Brown's interesting and ingenious steering-gear, in 
which the whole engine is made to move and then stops itself after turning the 
rudder through the required angle, see Engineering, Vol. XLIX, p. 491. 

f Constructor, §§ 319-332. 



CHAPTER XI. 



CHAINS INVOLVING SCREW MOTION. 



90. Formation of Screw Surfaces. — It has already been 
stated (§8) that lower pairs of elements can be con- 
structed in which the surfaces in contact are screws of uni- 
form pitch. Fig. 186 serves to illustrate the formation of 




Fig. 186. 

such surfaces. Imagine that a cyhnder A BCD is caused 

to rotate with uniform angular velocity, as indicated by the 

arrow, and let a cutting tool whose point is ground into the 

shape PQRS be moved with uniform linear velocity v in 

a direction parallel to the axis of the cylinder, so as to cut 

out a continuous groove in the material of the cylinder. 

If now the tool is so set that the lines PQ and SR when pro- 

276 



CHAINS INVOLyiNG SCREJV MOTION. 277 

duced pass through the axis of the cyhnder, the surfaces 
forming the side of the groove will be screw or helical sur- 
faces of uniform pitch. It will evidently be possible to 
form in a somewhat similar manner a hollow cylinder hav- 
ing the material of its inner surface removed in such a way 
as to leave a projecting thread of such a form as will exactly 
fit into the groove PQRS. The inner surface of this nut 
will be the exact counterpart of the outer surface of the 
screw, and when working together their relative motion 
must be a copy of the original relative motion of the cutting 
tool and the cylinder. In other words, the only possible 
relative motion of such a screw and its nut will be a motion 
of rotation, combined in a constant ratio with a motion of 
translation along the axis of rotation. By the term pitch 
we mean the distance (measured along the axis of rotation) 
through which the nut moves relatively to the screw during 
one complete relative rotation. Thus if oj be the angular 
velocity of the cylinder in radians per second, the time of 

one complete rotation will be — seconds. Durmg this time 

27tU 

the cutting tool will have moved a distance ; this expres- 

sion therefore gives the numerical value of the pitch. If we 
imagine that a piece of paper wrapped round the cylinder 
has the outline of the screw-thread marked upon it, and is 
then unwrapped, the line representing the edge of the screw- 
thread will be found to be straight, and it will make with the 
line representing the edge AB oi the cylinder an angle such 
as LMN. A little consideration will show that the tangent 
of this pitch-angle will be 

pitch of thread 
circumference of cylinder* 

It is quite easy to arrange a mechanism which will cut a 
screw-thread of variable pitch. This is, in fact, often done 
in rifling guns. In this case, if the angular velocity of the 
screw is uniform, the linear velocity of the tool must be 



278 



KINEMATICS OF MACHINES. 



variable, and the pitch-angle changes as we go along the 
thread. A hollow surface the exact counterpart of the 
screw would then only fit exactly in one position, and no 
relative motion of such a pair of surfaces would be possible. 
It is for this reason that a screw pair composed of rigid 
elements must consist of screw surfaces of uniform pitch. 
The section of the thread, as governed by the form of the 
cutting tool producing it, may be of any convenient form, 
and a number of standard threads are described in text- 
books on machine design. The reader should note that 
screws are often made with two, three, or a larger number 
of threads by cutting the required number of independent 
grooves on the cylinder. These threads may further be 
either right- or left-handed. The thread in Fig. 186 is 
right-handed; Fig. 187 shows a left-handed screw having 




Fig. 187. 
three threads. We shall see later that these multiple - 
threaded screws are of importance in screw mechanisms 
involving higher pairing, and we now consider certain cases 
in which lower pairing of screw surfaces is used in chains 
containing rigid links only. 

91. Screw Mechanisms Involving Lower Pairing of Rigid 
Links. — The relative motion of screw links is in general 
non-plane. On examination it will be found that in a screw 
and its nut, while there is at any instant rotation about the 
axis of the screw, there is also a simultaneous linear move- 
ment along that line. In more complex cases of the screw 



CHAINS myOLVING SCREIV MOTION. 



279 



motion of two bodies it has been pointed out * that there is 
at any instant a line common to the two bodies, called the 
twist axis, about and upon which each body is (at the instant 




/////////////////////////////////// 
Fig. 188. 

considered) turning and sliding relatively to the other body. 
In this work but little space can be devoted to the consider- 
ation of relative motion of this complex character, and in this 
section we shall discuss some of the simpler screw mechan- 
isms involving lower pairing. 

The simplest screw chain is shown in Fig. 188; it com- 
prises three pairs — a screw pair ah, a turning pair ac, and a 
sliding pair he. This chain is of common occurrence in 
the form of a screw press. By a suitable choice of the pitch 
of the screw we can obtain a machine in which a large 
angular motion of a gives us a comparatively small linear 
motion of 6, so that in a copying-press, for instance, a large 
pressure is obtained by applying a relatively small force to 
the end of the screw arm. 

If the screw has a sufficiently fine pitch this machine 
cannot be reversed ; that is, it is not possible by the applica- 
tion of an axial force to the nut h to cause rotation of a. 
By making the pitch of the screw sufficiently great, how- 
ever, this action becomes possible, as in the common Archi- 
medean drill. 

A little consideration will show the reader that we may 
look upon a sliding pair as a screw pair of infinite pitch, 
while a turning pair is also a special case of a screw pair in 
which the pitch is zero. Accordingly we may expect to 

* Kennedy, Mechanics of Machinery, § 68. 



28o 



KINEMATICS OF MACHINES, 



find mechanisms of three links containing two screw pairs 
and a sliding pair, or two screw pairs and a turning pair 
(as shown in Figs. 189a and 1896), the pair he or ac in Fig. 



m 



c-r,h'7?m 




Fig. 189^!. 

188 having been modified into a screw pair. Further, it is 
possible to transform the last remaining turning pair of 
Fig. 1896 into a screw pair and obtain a chain of three links 
and three screw pairs. The reader should have no difficulty 
in sketching for himself such a chain. 




Fig. 189^. 

Fig. 189a represents the chain containing two screw 
pairs and a sliding pair, as employed in a form of steering- 
gear; the complete gear forms a compound chain of seven 
links, four of which (shown by dotted lines) are added to 
the screw chain itself. The screw a, on which are cut two 
separate threads, right- and left-handed respectively, gears 
with two nuts h and c which evidently have a relative sliding 
motion, approaching or receding from each other as a 
rotates. The links / and g connect h and c to the arms of a 



CHAINS INVOLVING SCREIV MOTION. 



281 



yoke e secured to the rudder head. The frame or fixed Hnk 
d is the hull of the ship, to which are fixed the bearings in 
which a and e rotate. Plainly, rotation of a will cause the 
rudder to turn. 

Fig. 1896 shows the chain containing two screw pairs 
and a turning pair. Its most important application in 
practice will be discussed when we deal with screw chains 
involving fluid links. (See § 92.) 

A great variety of more complex screw chains are in 
practical use. Fig. 190 shows a crossed screw chain often 
employed as a portion of the reversing- gear of steam-engines. 
It consists of five links and contains a screw pair ab and 



eZERSING SHAFT 




Fig. 190. 

four turning pairs. The rotation of a hand-wheel on a 
moves the reversing-shaft from its position when the engine 
goes ahead to the position when the engine is in backward 
gear. 

The mechanism of Fig. 190 is a simple example of a class 
of mechanisms involving general screw motion, in which the 
relative motions of the links are often very complex. The 
majority of such chains, in fact, have not been worked out 
kinematically, but the more complicated general screw 
mechanisms find so small a field of usefulness that we shall 
not devote any space to them here. 



282 



KINEMATICS OF MACHIf^ES. 



92. Screw Mechanisms containing Fluid Links. — One of 

the simplest screw mechanisms containing a fluid Hnk is the 
rifled gun shown in longitudinal section in Fig. 191. This 
train consists essentially of three links. We have the gun 
itself, a, having traced upon the surface of its cylindrical 
bore the rifling, in the shape of a many-threaded hollow 
screw shown in cross-section SitAB. The projectile or shell 
b is introduced at the breech of the gun, which is closed by 
a screw-plug or breech-block, and the projectile is provided 
at its base with one or more copper driving-bands, which are 




SECTION ON A-B 



Fig. 191. 



of such a size that when the projectile is forced through the 
bore the projecting portions of the rifling cut into the copper, 
and in this way cause the shell to rotate. The third link, c, 
is of course the gas which is produced by the combustion of 
the charge, and which exerts the pressure necessary to propel 
the shell. It should be noted that the pitch of the rifling 
has to be large, compared with the calibre or diameter of the 
bore of the gun. Frequently the pitch of the rifling is not 
uniform, but is so designed as to decrease from the breech 



CHAINS INyOLyiNG SCREIV MOTION. 



283 



to the muzzle in such a way as to give as nearly as possible 
uniform angular acceleration to the shell. 

In Fig. 192 is represented a mechanism which is a special 
case of the screw-chain used for such important purposes 
as the propulsion of ships (screw propeller), the measure- 
ment of speed through fluid (anemometer, patent log), the 




Fig. 192. 
Utilization of the energy of the wind (windmill), and so on. 
The figure shows diagrammatically a screw conveyor used 
for the purpose of forcing broken coal from the hopper H 
into the furnace F of a mechanical stoker. We have here 
a rotating propeller a of peculiar form ; it is contained in a 
casing h, and acts on the powdery material enclosed by the 
casing. The reader will note that in the screw propeller of a 
ship we have exactly the same mechanism, except that the 
outer casing is used only under special circumstances,* and 
the material acted upon is fluid. 

The mechanism of Fig. 1896, when modified by the 
substitution of a fluid link for the piece h, takes the form of 
the parallel-flow (Jonval) turbine of Fig. 193, and is used 
for purposes of motive power. Here c, the turbine- casing, 
carries a bearing for a, the hollow shaft, and also has upon 
it a number of fixed guide-blades corresponding kinemati- 
cally to the hollow screw-thread of Fig. 1896. The fluid, 



*Barnaby, Marine Propellers, Chapter VII. 



284 



KINEMATICS OF MACHINES. 



lushing past these blades, encounters the blades of the 
turbine-wheel a, to which it communicates motion. The 
kinematic correspondence of the two mechanisms is evident. 
It should be noted that the surfaces of the guide-blades 
and buckets of a turbine, or of the blades of a propeller, 




-FIXED SUPPORT FOR SHAFT 
ROTATING HOLLOW SHAFT 



I ■ » \ \ \ *. 

V \ VlX^D cbilDE Bl^ADES \ 



' ' / / / / 

' ROtATING TURBINE/BLADES 
/' / / / / / 

' / ' / / / ' 



-^^13 




Fig. 193. 

are not necessarily true helical surfaces. With solid links 
we have seen that in order to obtain lower pairing the 
screw surfaces must have uniform pitch. The adoption of 
a varying pitch in the rifled gun is only possible because 
the copper driving-band, which pairs with the rifling, is nar- 
row and so soft as to be deformed with comparative ease. 
When we consider the pairing of fluid links with such sur- 
faces, however, the mobility of the fluid permits of great 
latitude in the form of the curved surface over which it flows. 
We have so far considered only screw-threads traced 
upon a cylinder, but there is no reason why such threads 
should not be formed on a conical surface, or indeed upon 
many surfaces of revolution. Fig. 194 shows a thread cut 



CHAINS INVOLyiNG SCREIV MOTION. 



285 



Upon a globoid, for instance, Let us now imagine a screw- 
thread traced upon a conical surface, as is the case in some 
forms of self -centring chuck,* or in the breech-blocks of 
certain quick-firing guns.j From such a screw-thread it is 




Fig. 194. 

but a step to the formation of such a surface as that of 
the vane of the wheel of a centrifugal pump (Fig. 176) or 
the vane of a radial-flow turbine, where the blades form 
what may be termed a screw surface projected on a plane. 
The kinematic chain of Fig. 176 is then really a modification 
of that shown in Fig. 193, the guide-blades being suppressed, 
and the whole forming a pump instead of a motor. The 
curves of the blades in a centrifugal pump are formed in 
such a fashion that their rotation impels the fluid from the 
centre to the outside of the pump-casing. They are thus 
spiral in form, or may even take the shape of radial straight 
lines. 

93. Screw-wheels and Worm-gearing. — In machine con- 
struction screws are employed not only in lower pairing for 
driving, or being driven by, rigid nuts, but also, in higher 
pairing, for gearing with rotating toothed wheels. In this 
case contact between the screw and the link with which it 



* " Horton " chuck. 



f Engineering, Vol. LXI. p. ii. 



286 



KINEMATICS OF MACHINES, 



pairs takes place either along a line or at a point. The 
ordinary worm and worm-wheel is the most familiar example 
of such gearing. Fig. 195 represents in plan and elevation 



c— 







A 










r5= 








H ' ■ 







■■ 


...... 


1 


K- 


- 


■ ■ 




, 


la 


""^-. 






















r 




B 


= 



— D 




C'_ 



M 




Fig. 195. 

two cylindrical wheels, a and h, whose axes AB and CD do 
not intersect and are at right angles in plan. The wheels 
are in contact at the point through which passes LO'M, 
the common perpendicular Xo AB and CD. The length of 
this common perpendicular is of course the sum of the 
radii of the two cylinders. Let a helical line or screw-thread 
be traced on the surface of a, so as to pass through the com- 
mon point 0, the pitch-angle of this helix being XOC, Also 



CHAINS INVOLVING SCREIV MOTION. 287 

suppose that a second helical line, not shown on the diagram, 
of pitch-angle XOA, is traced on the surface of h, so as also 
to pass through the point 0. The two helices will then 
touch at that point, and the line XOY will be their common 
tangent. If now the helix on h is replaced by a projecting 
thread, while that on a is converted into a corresponding 
groove into which the thread gears, any rotation of a about 
its axis AB will cause the rotation of h about its axis CD, 
and this relative motion of a and h will be continuous if we 
provide a series of projecting threads on h so spaced as to 
come into gear in succession with the thread or groove on a. 

It will be noted that, in Fig. 195, a is a single-thread screw, 
while the wheel 6 is a portion of a many-threaded screw, the 
number of threads on h being equal to the number of times 
that the pitch p is contained in the circumference of h. We 
can, however, evidently make pairs of screw-wheels in which 
a as well as 6 is a portion of a many-threaded screw, and a 
pair of such wheels is shown in Fig. 196, the teeth or threads 
being represented by the inclined lines. In speaking of the 
pitch of the teeth of these wheels, we must distinguish be- 
tween (i) the helical pitch, or pitch of the screw-thread {p in 
Fig. 195) ; (2) the normal pitch, or distance from centre to 
centre of teeth, measured at right angles to their length {q in 
Fig. 196) ; (3) the circumferential pitch (r, Fig. 196) ; (4) the 
axial pitch, or distance from centre to' centre of teeth meas- 
ured parallel to the axis of the wheel (5-, Fig. 196). A little 
consideration will show that in a pair of screw-wheels the 
circumferential pitch of each must be equal to the axial 
pitch of the other, supposing that, as in the figure, the axes 
of the wheels are at right angles in plan. 

We have now to find the angular velocity ratio of the 
wheels a and h. It is plain that since the teeth of a and 6, 
while the wheels rotate, remain in continuous contact, their 
velocity measured along a line drawn perpendicular to their 
common tangent at the point of contact and lying in the 
plane which touches both wheels must be equal. In Fig. 197 



288 



KINEMATICS OF MACHINES. 



this common velocity is represented by the line v^. Now 
let (o^ and Wj, be the angular velocities of a and h respectively, 



while r^ and r^ are their radii. 



Then if v^, v^ are the actual 



linear velocities of points on the pitch-circles of a and 6, we 
have 

'^a = ^a'^a ^nd v^ = co];r^. 

The lines OL and OM in the figure are supposed to be 




Fig. 196. 

drawn in the plane touching both wheels, and represent in 
magnitude and direction the linear velocities of the respect- 
ive pitch surfaces. ON is drawn in the same plane, and 
represents v^, the common velocity of the teeth of both wheels 
measured in a direction perpendicular to the common tan- 
gent of the teeth. 

The velocity v^ may be resolved into two components. 



CHAINS IN GOLFING SCREPV MOTION. 



289 



namely, ON, the velocity of a point on the tooth of a resolved 
in a direction norraal to the line of the tooth, and NL, the 
velocity resolved along the line of the tooth. Similarly OM 




Fig. 197. 

may be resolved into two components at right angles, ON 
and NAd] as we have seen, the component normal to the 
line of contact of the teeth must be the same for both wheels, 
because it represents the common velocity v^. The line LM 
will represent the relative linear velocity (along the tooth) 
of the pitch surface of the wheel a relatively to that of the 
wheel by or the speed with which the teeth slide lengthways 
over each other. 



290 KINEMATICS OF MACHINES. 

Now consider the circumferential and axial pitches of the 
two wheels. From the figure, by similar triangles 

v^ _ OL _ PR axial pitch of h 

Vjj OM QR circumferential pitch of b 
circumferential pitch of a 



circumferential pitch of b ' 



and 



v^ r^ _circumf. pitch of a nr^ 

X 



— = • 
io^ v^ ' r^ Ttr^ '^ circumf . pitch of b 



number of threads on b 
number of threads on a 

It is thus seen that in screw gearing of this kind the 
velocity ratio is independent of the sizes of the wheels, and 
depends solely on the number of threads with which they 
are provided. 

A particular form of screw-gearing is frequently em- 
ployed to transmit motion with a high velocity ratio 
between shafts at right angles in plan. The smaller wheel 
has only one, two, or three threads, of small axial but large 
circumferential pitch, and is known as a worm, while the 
worm-wheel has many teeth, of sma^ll circumferential but 
large axial pitch. The velocity ratio is, as we have just 
seen, simply the inverse ratio of the number of threads. 
Worm-wheels of good design have the form of their pitch 
surfaces modified so that their teeth are no longer screw- 
threads traced on a cylindrical surface, but are formed so 
as to obtain a larger area of contact between the teeth than 
would be possible in the case of a cylindrical screw-wheel.* 
The teeth of a pair of accurately formed cylindrical screw- 
wheels of rigid material would only touch in a point; in 
practice there would of course be a very small but percepti- 
ble area of contact. Such wheels are therefore most suit- 
able for light loads; and for heavy service, worm-gearing, 
in which the screw-thread and wheel-tooth may have line 

* See § 94. 



CHAINS INyOLyiNG SCREIV MOTION. 



291 



contact, is preferable. Fig. 198 shows the appearance of 
screw- gearing and worm-gearing as actually made. 

The axes of a pair of screw-wheels may of course make 
any desired angle in plan with one another. In a given 
case, when this angle, the sum of the radii of the pitch 
surfaces, and the velocity ratio have been decided, a number 
of different pairs of wheels may be designed which will obtain 




Fig. 198. 

the intended result ; the difference between them depending 
on the pitch-angles w^hich are selected. A numerical exam- 
ple will make this clearer. Suppose that screw-gearing is to 
be designed to connect two shafts making an angle of 60° 
in plan, the shortest distance between the axes being 10 
inches, and the velocity ratio three to one. Two solutions 
of this problem are shown in Fig. 199. In the' first case a 
pitch-angle of 60° has been chosen for each wheel ; in the 
second case the line of contact of the wheel-teeth is parallel 



292 KINEMATICS OF MACHINES.' 

to the axis of h, and the pitch-angles of the wheels a and h 
are respectively 30° and 90°. The wheel h is thus a spur- 
wheel. The velocity diagrams showing the relation between 
v^, Vf^, and v^ are drawn in the two cases. In order to deter- 
mine the radii of the pitch surfaces of the wheels, we have in 
the first case 

But ^=^^^ and -"=-. Hence 



The distance between the axes being 10 inches, plainly 
r„ = 2.5'' and r,- 7.5". 

In the second case v^ = 2V,; hence, since — =— as before, - 

^a 3 

r„ 12 

^b 3 3 

so that r^ = 4" and r^ = 6". 

In both cases the number of teeth on the wheels a and b 
must be in the ratio 1:3, but in the first case the wheels 
have the same circumferential pitch, and the radii are there- 
fore inversely as the angular velocities ; while in the second 
case the circumferential pitches of the wheels a and b are in 
the ratio 2 : i, so that the radii are in the proportion 2:3. 
From the diagrams it is evident that the sliding velocity of 
the teeth will be in the first case 

and in the second case 

It is noteworthy that for the same speeds of the shafts in 
the two cases v^ will be greater in the second instance than in 
the first in the proportion of 4 : 2 . 5 . Hence the sliding veloc- 
ity of the teeth will be greater in the second design in the pro- 
portion 2V s : 2.5, or nearly 1.39 : i. This is shown in the 



CHAINS INVOLyiNG SCREIV MOTION, 



293 



figure, where the velocity diagrams, shown in heavy Hnes, 
are both drawn to the same scale. In screw-wheels the 
sliding velocity of the teeth is a minimum when the wheels 
have the same pitch angle. 




PITCH ANGLE 

OF a=60° 

OF 6 = 60° 



AXIS OF 



PITCH ANGLE 

OF a = 30" 

OF 5 = 90° 



Fig. 199. 

94. Forms of Teeth in Screw-gearing and Worm-gearing, 

— The cylinders shown in the previous figures are of course 
intended to represent the pitch surfaces of the actual screw- 



294 KINEMATICS OF MACHINES. 

wheels. We have seen that the relative linear motion of 
these surfaces in the plane which touches them both is com- 
posed of a relative sliding motion along the line which is the 
common tangent to the teeth at the point of contact, and a 
common movement along a line perpendicular to that com- 
mon tangent. If we consider only this common velocity, 
it becomes plain that we have here a case similar in some 
respects to that of a pair of spur-wheels. In spur-gearing ' 
the teeth have a common velocity along a line which is per- 
pendicular to the line of centres and to the line of the teeth, 
and the relative motion is simply a rolling together, the 
virtual axis being a line parallel to the wheel axes and pass- 
ing through the pitch-point. In a pair of screw-wheels the 
relative motion is also one of rolling ; the virtual axis being 
the common tangent {XOY, Fig. 195), but this motion is 
combined with a relative sliding along the common tan- 
gent. This line may therefore be considered as the twist 
axis"^ of the two wheels, and it may be shown that the twist 
axodes of a pair of screw-wheels form a pair of hyperboloids ;t 
the two surfaces rolling together and at the same time slid- 
ing along their line of contact. If the axes of the screw- 
wheels are parallel, the hyperboloids become cylinders and 
there is no sliding motion along the line of contact. 

The above principles must be considered in determining 
the proper profiles for the teeth in screw-wheels. We must 
in fact imagine that the pair of pitch surfaces are cut by a 
plane passing through the point of contact and perpen- 
dicular to the common tangent. We must then take such 
shapes for the teeth as would work together correctly if 
formed on a pair of pitch-circles having the same radii of 
curvature as the sections — in which the actual pitch surfaces 
are cut by our imaginary plane — have at their point of con- 
tact. 

Evidently the traces of the cylindrical pitch surfaces 
of the screw-wheels on the imaginary plane will be a pair 

* See § 91. t See § 95. 



CHJINS INyOiyiNG SCREIV MOTION. 



295 



of ellipses, always touching at the ends of their minor axes. 
The plane will make with the median plane of each wheel 
an angle (90° — «f), where (x is the pitch-angle of the helical 
teeth or screw-threads, and the semi-minor axis of the ellipse 

T 

will be r; the semi-maior axis being -^^ , where r is the 

■' ^ sm « 

radius of the pitch surface of the wheel. In Fig. 200 is 
shown the pitch surface of a screw-wheel (radius r) having 
traced upon its surface a helix of pitch-angle XOC. As in 
Fig. 195, the line XOY is a tangent to the helix at the point 
O, and if were the point of contact of the wheel with an- 
other, XOY would represent the common tangent or line 
of contact of the teeth. 




B 
Fig. 200. 

xV plane passing through and perpendicular to XOY 
will cut the cylinder in the ellipse LMNO, of which the major 

2r 



and minor axes will evidently be 



sm « 



and 2r respectively. 



In order to find the proper form of tooth profile (taken of 



296 KINEMATICS OF MACHINES, 

course in the plane of LMNO and at right angles to the 
centre line of the tooth), we must take as an imaginary 
pitch -circle the circle of curvature of the ellipse at the point 
0. The radius of curvature of the ellipse at this point is 

easily shown to be . ^ ; hence m a pair of screw-wheels the 

^ sm^ « ^ 

teeth profiles should be designed as if they belonged to a 
pair of ordinary spur-wheels of radii 

and 



sin^ 01^ sin^ ol^ 

where a^ and a, are the pitch-angles of the teeth of the screw- 
wheels a and h. 

Screw-wheels are often used to connect shafts which are 
parallel. In this case the pitch-angles of the wheels are of 
course equal, and if the wheels are properly designed there 
is no difficulty in having two or more pairs of teeth in contact 
at once. In order to avoid the prejudicial effect of the end 
thrust developed when a pair of such wheels are doing heavy 
work, it is usual to make each wheel of two parts, similar 
in pitch, but one half right-handed and the other half left- 
handed. Fig. 201 shows such a double helical pinion of the 
form employed in a rolling-mill. The teeth profiles in such 
wheels, taken on a section by a plane perpendicular to the 
axes, will be the same as those required for spur-wheels of 
the same diameter and circumferential pitch of teeth; for 
in these wheels there is no sliding motion along the teeth. 
If properly shaped, the teeth will be in contact along short 
lines inclined more or less towards the pitch surfaces, accord- 
ing to the pitch-angle chosen for the helices. When this 
pitch-angle is 90° the wheels become spur-wheels, in which 
the lines of contact of the teeth are of course parallel to the 
pitch surfaces. 

When worm-gearing is constructed with cylindrical 
pitch surfaces, and teeth of uniform cross-section, contact, 
as in the case of other screw-wheels, occurs at a point only, 
and the forms of teeth may be found by the method just 



CHAINS INVOLVING SCREIV MOTION. 



297 



described. It is not difficult, however, by modifying the 
form of the wheel, to obtain worm-gearing having linear 
contact. The method of doing this is fully explained in 
works on Machine Design.* 

The section of the worm and wheel by a plane perpen- 
dicular to the axis of the wheel, and passing through the axis 
of the worm, is that of a rack in gear with a spur-wheel, and 
the form of the worm-thread and wheel-teeth in this plane 
may be drawn by the methods already discussed in §§ 
66 and 67. The trace of such a plane is shown in Fig. 202 





bd 




AXIS OF 
WORM WHEEL 



Fig. 201. 



Fig. 202. 



by the line AB ; the figure represents the section of the worm 
and wheel by a plane containing the axis of the wheel and 
perpendicular to the axis of the worm. The form of a sec- 
tion of the worm by a plane parallel to the axis of the worm, 
and perpendicular to the axis of the wheel, is next to be 
determined; the trace of such a plane on the plane of the 



* Unwin, '< Machine Design," Vol. I, § 234. 



298 



KINEMATICS OF MACHINES. 



figure is the line CD, and its intersection with the worm- 
thread will take the shape of a rack having curved and iin- 
symmetrical teeth. The form of worm-wheel tooth required 
to gear correctly with such teeth must then be found by the 
proper method of construction, and the shape determined is 
to be used for the section of the worm-wheel tooth cut by 
the plane CD. A number of such sections, found for planes 
at different distances from the median plane AB, will enable 
a practically correct wheel-pattern to be made. As a rule 
such wheels are machine-cut by being rotated in correct 
relation to a steel cutter or hob which is a duplicate of the 
worm to be used. 

The Hindley worm * has a screw-thread of varying section 
traced on a non-cylindrical pitch surface whose outline is 
an arc of the pitch-circle of the wheel. This form of tooth, 




Fig. 203. 

if correctly cut by means of a hob, has line contact: the 
teeth touching the threads at all points in the median plane. 

95. Hyperboloidal Wheels. — It is possible to construct 
wheels which will transmit motion between inclined non- 
intersecting axes, and which are so formed that their teeth 
are straight and have line contact. The pitch surfaces of 

* American Machinist, March 25, 1897. 



CHAINS INVOLVING SCREIV MOTION. 299 

such wheels are hyperboloids of revolution, as has already 
been stated. In Fig. 204 let AB and CD. be the axes of a 
pair of such wheels; the line of contact of their teeth is to 
be the line XOY, passing through a point on LOM the 
common perpendicular to AB and CD. In general XOY 
will be parallel neither to AB nor to CD. 

If now we imagine the line XOY to be rotated around 
^5 as an axis, while its position in relation to AB remains 
unaltered, XY will describe in space the hyperboloid a; 
and similarly, if we suppose the rotation to take place about 
CD, the hyperboloid h will be described. The two hyper- 
boloids will of course touch along the line XY, which is in 
fact their twist axis when relative motion occurs. The 
smallest circular sections of the hyperboloids are known as 
the gorge-circles. We proceed to determine the relative 
angular velocity of a pair of such hyperboloidal surfaces, 
supposing that they roll together. It is to be particularly 
noted that hyperboloidal wheels differ from the cylindrical 
screw-wheels hitherto discussed, in that the pitch surfaces 
of the latter can touch only at a point, while those of the 
former are in contact along a Ime. The relative motions 
of the two kinds of wheels are, however, of the same kind, 
namely, a rolling together, combined with relative sliding 
along the line of the teeth. Let d^, 6^ be the angles (in plan) 
made by the projection of the line of contact XY with the 
projection of the axis of a and the projection of the axis of 
h respectively. The angular velocity-ratio of the wheels a 
and h must evidently be the same as that for a pair of screw- 
wheels of the same size as the gorge-circles of the hyper- 
boloids and having the same obliquity of teeth. The 
velocity diagram will therefore be that drawn in thick lines, 
by the method of § 93, and we shall have 

^ _ ^ n _ n-^'c COS ^2 _ ^fc cos e^ 

<^b ^a''^b cos^^- r^v^ r^cosd,' 
Now consider any point Y on the line of contact XOY. 
The normal to the curved surface of the hyperboloid at any 



300 



KINEMATICS OF MACHINES, 



point must pass through the axis; hence a straight line 
drawn through Y and normal to the curved surfaces which 




Fig. 204. 

touch there must intersect both CD and AB. Such a com- 
mon normal is shown in the figure, where RYS is its pro- 



CHAINS INVOLVING SCREIV MOTION. 301 

jection on a plane parallel to both axes AB and CD, and 
R'Y'M is its projection on a plane perpendicular to the axis 
oi h. In this view of course the points D, M, C, and S 
coincide. 

By a well-known principle of projection the real lengths 
of the segments of the common normal are proportional to 
their projections RY , YS, and RY\ Y'M. Hence 

r. LO RY' RY 



and 



r, OM Y'M YS 
co^~' YS cos d. 



2 



oj^ RY ' cos d^' 
Again, since the common normal is perpendicular to 
XOY, the line of contact, its projection RYS will be per- 
pendicular to the projection of XOY on a parallel plane; 
so that in the lower figure OYS and OYR are right angles. 
Thus, finally, 

oj„ DY 



CO, BY' 



This shows that the angular velocities of a pair of hyper- 
boloidal wheels are to each other in the inverse ratio of the 
lengths of the projections of the perpendiculars drawn from 
any point on the line of contact to the axes; these projec- 
tions being upon a plane parallel to both axes and to the line 
of contact. 

It should be noted, that in designing hyperboloidal wheels, 
if the angle between the axes (in plan) and the velocity ratio 
are given, the position of the line of contact (in plan) is de- 
termined. Thus in drawing such a pair of wheels we proceed 
as follows : 

(i) Draw the axes in elevation and in plan. 

(2) The velocity ratio being given, draw the line of con- 
tact XOY (in plan), determining the point Y by marking 
off Dy and BY having lengths in the proper ratio. 

(3) Draw SYR perpendicular to XOY, and also draw 
MY'R\ the projection of SYR on a plane perpendicular to 
the axis of h. 



302 



KINEMATICS OF MACHINES. 



(4) Through V draw X'O'Y' ; this detemiines the values 
of r^ and r j, and settles the sizes of the gorge-circles. 

(5) Proceed to complete the projections of the hyper- 
boloids, as shown. 

As in the case of screw-wheels, the velocity diagram 
shows the rate at which the teeth of a and h slide along each 
other. This relative sliding velocity is shown as v^ in Fig. 
204. 

It is not necessary in practice to use more than a com- 
paratively small portion of the hyperboloid for a working 
wheel. Fig. 205 shows a pair of hyperboloidal rollers, and 

a pair of skew-bevel wheels having 
the same velocity ratio. It will be 
seen that these bevel wheels corre- 
spond in fact to the end portions of 
the hyperboloids. The forms of the 
teeth of hyperboloidal wheels may 
be conceived as being marked out 
upon the cones whose surfaces are 
normal to those of the hyperboloids 
at the point of contact considered. 
Methods of doing this have been 
discussed by WiUis,* Rankine,t and 
others. Here it will be sufficient to 
note that the teeth of such wheels 
will not be of uniform section 
throughout their length. In the 
comparatively narrow hyperboloidal wheels generally used 
there is but little variation in the form of the tooth in 
passing from one end of the wheel to the other. An ap- 
proximately correct form of tooth may be determined for 
such wheels in the same way as for screw-wheels. 

In Fig. 204 for example, we may imagine the two hyper- 
boloids cut by a plane parallel to RYS and perpendicular 




Fig. 205. 



* Principles of Mechanism, p. 151. f Machinery and Millwork, p. 146. 



CHAINS INVOLVING SCREIV MOTION. 303 

to XOY. When the resulting sections are drawn out their 
circles of curvature may be approximately found, and the 
tooth-forms designed in the ordinary manner, remembering 
that the circumiferential pitch, and therefore also the normal 
pitch, increases as we pass from the gorge to the ends of the 
wheels. It will be seen that this method is practically the 
same as that adopted in the case of ordinary bevel-wheels 
(see § 98), and is equivalent to drawing out the teeth on 
the development of the cones previously mentioned. 

The subject of hyperboloidal wheels is treated at con- 
siderable length in MacCord's ' ' Mechanical Movements," to 
which work the reader is referred for further information. 



CHAPTER XII. 
SPHERIC MOTION. 

96. Spheric Motion in General. — Spheric motion has 
already been defined in § 6, and it has been explained 
that in such motion any given point in the moving body 
remains on the surface of a sphere described about a certain 
fixed point as centre. Two bodies having relative spheric 
motion will therefore have this point as a common centre. 

We can stud}^ the relative motion of two or more such 
bodies by imagining that they are cut by a sphere described 
about the common point as centre, and we can then con- 
sider the movement of these spheric sections exactly as we 
considered the motion of the plane sections or projections 
of bodies having plane motion. Plane motion may indeed 
be looked upon as a particular case of spheric motion where 
the radius of the sphere is infinitely great. 

We may therefore suppose that propositions proved with 
regard to plane motion will hold good, with certain necessary 
modifications, with regard to spheric motion also. It will be 
convenient, first of all, to consider the motion of a spheric 
figure on the sphere of motion, just as we considered in 
§ 5 the motion of a plane figure in the plane of motion. The 
position of the spheric figure will of course be defined if we 
know the position of two of its points. 

In Fig. 206 (a) a figure on the surface of a sphere LMN 
has the positions of two of its points {A and B) defined. 
Let the figure, which represents a body having spheric 
motion, be moved from a position AB to ^, new position, 
A^B^\ the movement being executed in a very small period 

304 



SPHERIC MOTION. 



305 



of time, and being therefore an exceedingly small displace- 
ment. The paths of the points A and B will then practically 
coincide with portions of great circles passing through A 
and A^, and B and B^, respectively. Now let arcs of great 
circles be drawn passing through L and M, the middle points 
oi AA^ and BB^\ let the planes of these great circles be 
respectively perpendicular to those of the great circles ALA^ 
and BMB^, and let them intersect at A^. Draw ON passing 




Fig. 206. 

through 0, the centre of the sphere. It is then evident that 
the actual small displacement of the body AB is the same 
as if it had undergone a rotation about the axis ON, for N 
is the point on the surface of the sphere at which AA^ and 
BB^ subtend equal spherical angles. This follows from the 



3o6 KINEMATICS OF MACHINES. 

fact that the spherical triangles ANB and AJSIB^ are equal 
in all respects. 

It may happen that L and M both lie on the same great, 
circle, as in Fig. 206(b), in which case our construction fails. 
The point A^ is now to be taken at the intersection of the 
great circles AB and A^B^, and it is evident, as before, that 
the angle subtended at A^ by the arcs AA^ and BB^ is the 
same, and that ON is the axis of rotation. Any actual 
motion oi AB on the surface of the sphere may be considered 
as being made up of a series of infinitely small displacements, 
to each of which there corresponds one position of the axis. 
ON. ON^ is therefore the vi7tual axis of the motion of AB 
with regard to the sphere. The reader should compare the 
foregoing argument with that in § 5 applying to plane motion. 

We may call the surface described by ON in the sphere 
the axode oi AB with respect to the sphere. Two bodies, a 
and b, having relative spheric motion will of course have a 
pair of such axodes ; the axode of a being imagined as being 
described in the body b and vice versa, just as in the case 
of plane motion: and, further, this relative motion may 
be represented by the rolling together of such axodes. 

Perhaps an example may make this clearer. Fig. 207 
represents the pitch surfaces of a pair of bevel- wheels ; their 
axes intersecting at 0. These wheels are intended to trans- 
mit angular motion uniformly between shafts whose axes 
intersect, and their motion will evidently be exactly the 
same as that of a pair of circular cones of corresponding 
shape rolling together without slipping, and having a com- 
mon apex at 0. A pair of such cones, a and b, and their 
frame, c, will have relative spheric motion about the point 
0. The lines OA^^ and OA^^^ are of course the axes of a 
and b with respect to the frame c; the line OA^^ along which 
the cones are in contact is the virtual axis of a with respect 
to b, and the surface of each cone is therefore the axode of 
the other. (Compare the relation between the pitch sur- 
faces in spur-gearing.) 



SPHERIC MOTION, 



307 



It is evident that in Fig. 207 the three virtual axes of 
the three moving bodies a, h, and c are in one plane; we 
proceed to show that this is true of any three bodies having 
relative spheric motion. The figure already used to illus- 




FiG. 207. 

trate the corresponding proposition for plane motion is 
repeated here. 

Let the bodies be a, h, and c ; there must be some point 
in space, 0, which is common to the three bodies, and 
through which, therefore, their three virtual axes must 
always pass. Let the paper (in Fig. 208) represent the pro- 



3o8 KINEMATICS OF MACHINES. 

jection of part of a spherical surface, and let 0^^, 0^^, and 
0^^ represent the traces on this surface of the virtual axes 
OA^^, OA^^, and 0.4^^, respectively. Then, following the 
reasoning of § 5, we may say that the line OA^^ belongs, 
for the instant considered, to both a and c. As a line 




Fig. 2o8. 

in a it is turning relatively to b about the line 0.4^^, and is 
therefore moving in a direction perpendicular to the plane 
containing OA^^ and OA^^. As a line in c it must similarly 
be moving in a plane perpendicular to the plane contain- 
ing OA^^ and OA.^^. The line OA^^ is therefore moving in a 
plane normal to each of two planes which contain it, and 
these two planes must coincide. The three lines OA^j^, 
OA^^, and OAj^^ thus lie in one plane. 

97. Spheric Mechanisms having Lower Pairing. The 
Conic Quadric Crank-chain. — It is not difficult to devise 
micchanisms corresponding to the plane mechanisms of 
Chapter III and IV, but having spheric motion of the va- 
rious links. To do this it is only necessary to arrange that 
the axes of all turning pairs meet in a point instead of being 
parallel, and that the lines of motion of sliding pairs follow 
great circles on the surface of the sphere of motion. 

The axodes of the links of such mechanisms will, as we 



SPHERIC MOTION. 



309 



have seen, be conical surfaces (not necessarily circular 
cones), and the mechanisms are therefore called by Reu- 
leaux conic chains. A model representing a conic quadric 
crank-chain is shown in Fig. 209, and it may be remarked 
that, as in the case of plane mechanisms, the actual form of 
the links is unimportant from a kinematic point of view so 
long as the axes of the elements are in the correct position 
and the links do not foul one another during motion. The 
reader should compare the chain of Fig. 209 w^ith that of 
Fig. 40. 

In studying conic mechanisms we may note that, instead 
of considering the actual lengths of the various links, we have 




Fig. 209. 

now to deal with the angles subtended by those links at the 
centre of the sphere. The relation between a plane mech- 
anism and the correspondmg spheric chain is in this respect 
like the relation between a plane triangle and a spherical 
one. In order to connect the elements of two turning pairs 
making an angle a with one another, we can thus use either 
a link subtending the angle a or one subtending 180° - a. 



3IO KINEMATICS OF MACHINES. 

Such a substitution may of course totally alter the appear- 
ance of the mechanism, but will produce no kinematic 
change. 

The reader will recall the way in which we imagined the 
plane or cylindric quadric crank- chain of Fig. 60 to have 
one of its pairs transformed from a turning to a sliding pair. 
This change involved the expansion of the elements of the 
turning pair cd until the radius of their working surfaces 
became infinitely great. In Fig. 210 is illustrated the cor- 
responding alteration by which a conic quadric crank-chain 
becomes a conic slider-crank chain. The chain of Fig. 210 (6) 
is a conic lever-crank chain, in which the angle of the links 
a and h is in each case 90°. It will be evident that by in- 
creasing the radius of the turning pair ah until it is equal 
to the radius of the sphere we get the chain of Fig. 210 (c), 
having exactly the same relative motions as before, but 
having as the link h a block sliding in a groove formed in a 
and following a great circle on the sphere. We may term 
this mechanism a spheric or conic slider-crank. The right- 
angle links in the conic chain correspond to the infinite links 
in the plane mechanism. The sketch Fig. 210 (a) represents 
a conic quadric crank-chain in which 6 = 45°. If a some- 
what similar transformation were carried out in this case 
we should obtain a crossed conic slider-crank chain, the 
groove on the surface of the sphere following the trace of 
the axis OA^^, which in this case is not a great circle and 
no longer passes through A ^^. 

As an exercise the reader should endeavor to devise for 
himself spheric mechanisms corresponding to other simple 
plane mechanisms. Comparatively few conic crank-trains 
have found application in practice,* and in general their 
industrial importance is not very great. We shall proceed 
to follow somewhat more closely the action of the conic 
quadric crank-chain, which is utilized in the form of the 

* For a discussion of one of these, the Tower Spherical Engine, a conic 
chamber quadric crank-chain, see Kennedy, Mechanics of Machinery. § 65. 



SPHERIC MOTION. 



3" 



/ 



\ 



\. 




y^ 



/ 



1—1 
o 



\ 



\. 




/ 



si 




312 



KINEMATICS OF MACHINES. 



well-known Hooke's or Universal Joint, for connecting 
shafts whose axes are not parallel and meet in a point. 

One arrangement of this mechanism is shown in Fig. 




Fig. 2iia. 

2iia, together with a diagram, Fig. 21 16, showing the vari- 
ous links drawn on the surface of a sphere, after the fashion 




Fig. 211^. 



of Fig. 2 10 (a). The plane of the paper is supposed to be that 
of the axes OA^^ and OA^^, b and d being the links corre- 
sponding to the two shafts, while a is the fixed link, and c 



SPHERIC MOTION. 313 

connects b and d. We wish to find, for any given position 
of the mechanism, the angular velocity ratio of b and d. 
The hnks b, c, and d each subtend an angle of 90° at the 
centre of the sphere, while a subtends an angle equal to 
180° — (angle between axes of shafts). 

It may be noted that while b and d will obviously per- 
form successive quarter-revolutions in equal times, their 
angular velocities are not necessarily equal in any given 
position. The angular velocities will in fact be equal at only 
four points in each revolution. 

Utilizing the proposition of § 96, we can easily find the 
virtual axes of the raechanism of Figs. 211a and 211b. Thus 
OA ^^ must lie in the plane containing OA^f^ and OA ^^ ; it must 
also lie in that containing OA^^ and OA^^. Similarly the 
point A^^ is found at the intersection of the great circles 
passing through A^^A^^ and A^^A^^. In the case of spheric 
motion it is important to remember that the virtual axis of 
two moving bodies is a line which is common for the instant 
to the two bodies, and which has the same angular velocity 
whether it is regarded as belonging to one or to the other. 
This statement corresponds in the case of plane motion to 
the definition of a virtual centre as that point in the pro- 
jection of the two bodies on the plane of motion which is 
for the instant common to both projections, and has the 
same linear motion whether it is considered as a point in one 
body or as a point in the other. 

Having found the virtual axes of the mechanism, the 
relative angular velocities of the links can be determined 
graphically. In Fig. 2 116, for example, suppose that V 
is the linear velocity of the point P{A^^) in a direction tan- 
gential to the great circle A^^A^.^ and normal to the plane 
of the paper. The angular velocity of P (considered as a 

V 
point in b) about the axis OA^^ is of course -pinf, and this 

must be equal to the angular velocity of the link b with 
respect to the fixed link a. Similarly, since P is also a point 



314 KINEMATICS OF MACHINES, 

in d, the angular velocity of d must be -p^, where PM and 

PN are the perpendiculars dropped from P on to OA^^ and 
OA^^ respectively. Thus it follows that 



da 



It will be seen that the determination of the angular 
velocity ratio in this way involves the finding of the axis 
OA^^^, which can only be done by drawing the plane pro- 
jection of the whole mechanism in each position for which 
the velocity ratio is required. Since each such projection 
generally involves drawing three ellipses, the process is not 
very convenient, except in the cases where the plane of the 
link h makes an. angle of 90° with, or coincides in position 
with, the plane of the link a. The latter plane of course 
is the plane which contains OA^^ and OA^^. 

The positions mentioned are shown in Fig. 212. Let 6 
be the angle between the axes of h and d, let a be the angle 
made by the plane of h with a plane normal to the plane of 
the paper and containing OA^^, and let ^ be the angle be- 
tween the plane of <i and that of the paper. As before, the 
plane pf the paper contains OA^^^ and OA^^. It is then 
evident from the figure that when a = o, ^ = o, and 



^., PN 



<u^, PM cos d' 

Similarly when a = - and 5 = -. we have 

22 

—^ = =cos d 

aa 

The value of the angular velocit}'' ratio may also be 
found in another way. Fig. 213 shows three views of the 
mechanism, namely, a projection on the plane containing 
OA^j, and OA^^, a projection on a plane perpendicular to 
OA^^y and one on a plane perpendicular to OA^^. In the 



SPHERIC MOTION. 



315 




Fig. 212, 



. s: 



PROJECTION ON PLANE 
HORMAL TO AXIS OF d 




PROJECTION ON PLANE CONTAINING 
AX^,S.OF b AND d 



NORMAL TO AXIS OF 6 



Fig. 213. 



3i6 KINEMATICS OF MACHINES. 

latter view the ellipse A'D'C represents the projection of 
the path of the point 5 (the join of the links c and d), while 
the circle A'B'C represents the projection of the path of R, 
the join of c and h. If now we draw the angle A'OR = ^, 
so that OR' is the projection of the link 6, the correspond- 
ing projection of the line OS will be OS' . The real angle 
ROS is of course a right angle, being equal to the angle 
subtended by the link c ; and since the line OR' lies in the 
plane of the paper, the projection OS' must, by a well- 
known principle of projection, be at right angles to OR' , 
Hence it follows that the angle B'OS' = a =AOR'. Similarly 
it may be shown that in the other view D"OS" =^=A"OR". 
It is plain from the figure that 5' F' =^S"V" = perpendicular 
distance of 5 from plane of axes. Also OV = OS cos 6. Thus 

S"V" S'V _S'V' 
tan^— Qyr/ — Q^ ~ OV ^^^ 

= tan a cos d. 

From this expression, being constant, we get a relation 
between « and /?. By differentiation with regard to time 

seC^^ -J- = cos sec a-Tr- 

rry^ r . da ^ dB 

Therefore smce -^ =oj^^ and -^=oj^^, 

^ba _ i+tan^/? 
w^^ sector cos d ' 

But tan /?=tan a cos 6. Hence 

(0.„ cos^« . - 
-^= ^-hsm'arcos^ 

I — sin^o' sin^ d 
cos d 

Similarly it may be shown that 

Wfca cos d 



co^^ I — cos^/? sin^ ^ 



SPHERIC MOTION. 317 

The velocity ratio plainly has its maximum value! ^ ) 

when a=^ = o, n, 2 7r, etc., and its minimum value (cos ^) 

when a = B = -, — , — , etc. ; results which agree with those 
222 

previously obtained. 

In order to find the positions in which the shafts have 

the same angular velocity we have only to put -^ = i and 

da 

I — sin^ a=cos 6(1 — sin^ a cos 6) 

. , I — cos d 

or sm^ a = ^-x-. 

I — cos^ a 

For example, if ^ ==30° and cos ^ =0.86602, we have 

. , 0.13398 

0.25001 ^^^^ 
sin a= ±0.73205. 

The two shafts will then be moving with the same angular 
velocity when a = 47° 4', 132° 36', 227° 4', or 312° 36'; that 
is to say, four times in each revolution. 

It is evident from the relations thus obtained that if 
we connect two shafts by means of an intermediate piece 
and two similar universal joints, as is done, for example, in 
the feed gear of certain milling machines, then if the shafts 
are parallel so that 6^ = —0^, we have a^ =^,^, and the shafts 
will have uniform velocity ratio; the inequality of motion 
caused by the first universal joint being exactly compen- 
sated by the second. 

The method of studying the action of the conic quadric 
crank- chain may serve as an example of the way in which 
other conic mechanisms having lower pairing may be treated, 

98. Spheric Mechanisms having Higher Pairing. Bevel- 
gear. — Each of the spheric mechanisms discussed in the 
preceding section is the representative of a plane mechanism, 
the essential difference being that in a spheric mechanism 



3i8 



KINEMATICS OF MACHINES, 



the axes meet in a point instead of being parallel, and the 
relative motion of the links is spheric instead of plane. 

If we consider in a similar manner the change which 
would take place if the axes in spur gearing were made to 
intersect instead of being parallel, it is plain that the cylin- 
drical pitch surfaces would become cones, whose apices 
would lie at the point of intersection of the axes. The 
toothed wheels, whose relative motion corresponds to the 
rolling together of such conical pitch surfaces, are known 
as bevel-wheels, and such relative motion is, of course, 
spheric motion, as was shown in § 96. 

If we go a step further and imagine that the axes are 
not parallel and do not intersect, then the pitch surfaces 
become hyperboloids and the relative motion is screw 
motion, a state of things which has already been considered 
in Chapter XI. 




Fig. 214. 

Two right circular cones, a and b, whose axes are AQ 
and BQ, are in contact as represented in Fig. 214. If these 
cones are so rotated that there is no relative slipping at one 
point of contact, L, and if the point Q is the common apex 
of both cones, then at every other point of contact there will 



SPHERIC MOTION. 319 

also be rolling, without any sliding motion. Let the plane 
of the paper contain the axes QA and QB and the line of 
contact QL] then, if there is no slipping at L, and if oj^^ 
and io^^ are the angular velocities of a and 6, relatively to c, 
the fixed link (not shown in the figure) , we shall have 

(o„. F LN 



co,^ LM • V^ 



where V^ is the common linear velocity of the two cones 
at L, measured, of course, in a direction normal to the 
plane of the paper. Hence 






Further, if there is to be no slipping at another point of con- 
tact, /, we must have 



(O 



ac 



In 



^bc ^^ ' 



In LN 
a relation which shows that when -. — = ^^^r-j- the circular 

Im LM 

sections at In and Im will roll together if the sections at LN 

and LM do so. We may note that 

co^^ ^ sin BQL 
^hc sinAQL* 

It is easy therefore to lay out the pitch surfaces for a pair 
of bevel -wheels having any desired velocity ratio. We have 
only to arrange two cones having a common apex, and hav- 
ing a line of contact such that the lengths of the perpen- 
diculars dropped from any point on it to the axes are in- 
versely as the angular velocities. The cones need not 
necessarily have external contact. Bevel-wheels having 
internal contact can be made ; an internal bevel- wheel cor- 
responds to an annular spur-wheel. 

In practice frustra of the pitch-cones are used for the 



320 KINEMATICS OF MACHINES, 

pitch surfaces of bevel- wheels, and slipping is prevented by 
forming teeth on these pitch surfaces, exactly as in the case 
of spur-wheels, with the important difference that in the 
case of bevel-wheels the teeth are not of uniform section, 
but taper in such a fashion that they would vanish at the 
apex of the pitch-cone if they were continued to that point.* 
In Fig. 214 the virtual axes are, of course, QA, QL, and QB, 
QL being the virtual axis of a with regard to b. The third 
(fixed) link of the train is not shown. 

In Chapter VII, when studying the formation of the 
tooth profiles for spur gearing, we considered these as being 
plane curves, described upon a plane normal to the axes 
of the wheels and (in the case of cycloidal teeth) generated 
by the rolling together of the describing circles and the 
pitch-circles. It would perhaps have been more correct 
if we had considered the tooth surfaces as being generated 
by the rolling together of pitch surfaces and describing 
cylinders. In the case of bevel gearing the corresponding 
problem is more difficult, because it is necessary to picture 
in the mind the working faces of the teeth as being described 
by the rolling together of conical, instead of cylindrical, 
surfaces. Drawings connected with bevel-gear in general 
will require at least two projections, as we found when con- 
sidering hyperboloidal gearing. 

It is possible to assume the form of the working surface 
of the tooth of one bevel- wheel, and then to devise the form 
of the corresponding tooth of another wheel gearing with 
the first with uniform velocity ratio. The necessary and 
sufiicient condition for such uniformity will be that in any 
spheric section of the pair of teeth, drawn with the inter- 
section of the axes as centre, a great circle drawn to cut the 
tooth profiles at right angles at their point of contact will 
pass through the virtual axis of the pair of wheels. (Com- 
pare the corresponding condition for plane motion.) It is 
proper to employ spherical curves as forms for the teeth of 
bevel- wheels, these curves being drawn in similar fashion 

* See Fig. 207. 



SPHERIC MOTION, 



321 



to the plane involute or cycloidal curves used in the case of 
plane motion, and we shall now discuss the way in which we 
may imagine involute bevel- wheel teeth profiles to be gen- 
erated. In Fig. 215 the curves MSC, NTD represent the 




traces, on a spherical surface whose centre is 0, of two right 
circular cones, having axes OP and OQ. The great circle 
APRQB lies in the plane containing OP and OQ. A plane 
touching the two cones intersects the spherical surface in 
MRN, which is, of course, an arc of a great circle. If now 
we suppose that the plane surface OMN takes the form 
of a flexible sheet, it will be seen that a rotation of the two 
cones in the sense of the arrows would cause the sheet to 
wrap itself around the cone OSM and unwrap from the cone 
OTN, a point L on the edge of the sheet thus describing on 
the surface of the sphere the great circle NLM. With 
reference to the small circle SM, the point L will describe 
on the surface of the sphere the curve SL, which may be 
termed a spherical involute; and in the same way with 
regard to TN, the involute TL will be drawn, the two in- 



32 2 KINEMATICS OF MACHINES. 

volutes, of course, always touching at L. The line OL will 
thus describe on the cone OSM the ruled surface OSL, and 
the somewhat similar surface OTL will be generated by the 
relative motion of OL and the cone OTN. These surfaces 
will roll and slide together, always being in contact along 
such a line as OL, if the cones rotate with uniform velocity 
ratio ; and they can therefore be used as the working faces 
of the teeth of a pair of bevel- wheels whose axes are OP and 
OQ. The pitch surfaces of these wheels will be the cones 
OAR, OBR, which are indicated by dotted lines in the 
figure and touch along the line OR. A plane drawn through 
OL perpendicular to the plane OMN would be the common 
tangent plane to the tooth surfaces through their line of 
contact. The reader should compare this discussion with 
that of § 66, noting the modifications rendered necessary 
in adapting the reasoning to the case of spheric motion. 
He should also endeavor to work out for himself the method 
of forming cycloidal teeth for bevel-wheels by a method cor- 
responding to that used in § 67 for spur-wheels, remembering 
that the figure must be supposed to be drawn upon the 
surface of a sphere instead of upon a plane surface. 

In practice it is not convenient to use spherical sur- 
faces for drawing; in setting out the teeth of bevel-wheels 
it is therefore necessary to adopt a somewhat different 
method (due to Tredgold) which gives results closely ap- 
proximating to the truth. 

In Fig. 216, OAR, OBR represent the pitch surfaces of 
a pair of bevel- wheels, projected on a plane containing the 
axes OP, OQ. The arc APRQB is the trace of a spherical 
surface drawn with centre 0, gn which surface the outlines 
of the teeth should properly be described. For this surface 
we substitute the developable surfaces of the two cones 
XAR, YBR which touch the spherical surface along the 
pitch-circles AR, BR of the bevel- wheels. It is evident 
that these cones are in fact the same as the cones men- 
tioned (in the case of hyperboloidal wheels) in §95, 



SPHERIC MOTION. 



323 



and their curved surfaces are normal to the pitch surfaces 
of the wheels. The arcs RV and RW, drawn with X and 
Y as centres respectively, are in fact the developments of 




Fig- 216. 

portions of the pitch-circles AR, BR, and it is on these lines 
as pitch-circles that the tooth profiles are to be drawn.* No 
considerable error is introduced by this construction so long 
as the bevel- wheel has more than 24 teeth. 

In recent years a number of machines have been de- 
signed for the purpose of cutting bevel- wheel teeth f ; one 
of the most important and interesting from a kinematic point 
of view is that devised by Hugo Bilgram X for cutting teeth 
of the involute form. In this machine the correct form of 



* See Unwin, Machine Design, Vol. I. § 208. 
f Trans. Am. Soc. M. E., Vol. XXII. p. 672. 

\ See Engineering, Vol. XL. p. 21; also Journal of the Franklin Inst., Aug., 
1886. 



324 



KINEMATICS OF MACHINES, 



tooth is generated by the relative motion of a V-shaped 
cutter, representing the involute tooth of an imaginary 
plane bevel-wheel or crown-wheel, and of the wheel blank. 
The latter is rotated, during the formation of each face of 
each tooth, about its proper virtual axis. The meaning 
of the term plane bevel- wheel will be seen from Fig. 217. 




Fig. 217. 



In this figuue the pitch surface of a is a cone whose vertical 
angle is 2( oc\ where a is the angle between the axes. 

The pitch surface of h then has a vertical angle of 180° and 
is a plane. From the figure we see that 






With the same angle between the axes we might also have 
a wheel 6' of vertical angle 2la — 



L2 J 

In this case we should have a velocity ratio 



— a ] gearmg with a. 



SPHERIC MOTION. Z^S 




COS OL 



1 — 2 COS^ oc 



If in Fig. 2 1 7 we suppose the pitch surface of the plane 
wheel h to be fixed, and the wheel a to be rolled upon it, 
it is plain that a cutting-tool reciprocating along a diameter 
of the pitch surface of h will in successive cuts, if properly 
adjusted, fonn a tooth space in the wheel blank corre- 
sponding to a. It is on this principle that the action of the 
Bilgram machine depends. 

In another class of bevel-gear-cutting machines we may 
place those using a master gear or template. The mode of 
operation of the Rice gear-cutting machine, which is of this 
type, is illustrated in a diagrammatic fashion in Fig. 218; 
the details of the actual machine being somewhat differ- 
ently arranged. The blank A, from which the bevel- wheel 
is to be cut, has the tooth spaces roughly gashed out, and is 
mounted on a shaft to which is secured a template or mas- 
ter wheel B, having teeth of the correct profile formed upon 
it. There may be only one of these profiles, as in the sketch, 
or B may take the form of a complete wheel. The shaft 
carrying A and B can rotate about an axis OX, with reference 
to the frame D, which in its turn can be rotated about a 
vertical axis OY. The actual motion of the blank may 
therefore be any rotation compounded of movements about 
the axes OX and OY. The fixed base of the machine, E, 
carries an arm E^, in connection with which a rotating 
cutter C and a guide-plate F are so arranged that the face 
of the cutter and the face of the guide-plate lie in a plane 
containing the axis OY. The cutter having slightly entered 
the gash cut for the tooth space, the frame D is rotated 
about OY, and if the tooth form is kept in contact with the 



326 



KINEMATICS OF MACHINES. 



plane face of F, it is evident that the relative motion of F 
and B will be copied on a reduced scale by C and A , and the 
cutter will therefore form one side of one tooth of the blank 
wheel. With reference to the fixed link E, the blank and 
template are at any instant really rotating about some such 
axis as OR. In order to cut the opposite side of the tooth, 
the stop F must be moved parallel to itself by an amount 




Fig, 2i8. 
equal to its own thickness, and when brought in contact 
with the opposite face of the template a repetition of the 
same movement will cause the desired form to be produced 
on the other side of the tooth of the blank. One correctly 
formed template is thus made to serve for cutting a num- 
ber of bevel wheels. 

It is possible to construct bevel-wheels with spirally 
formed teeth, as in Fig. 219. These wheels differ from the 



SPHERIC MOTION. 



327 



skew-bevel wheels of Fig. 205 in that the axes of the wheels 
intersect, so that the pitch surfaces are of course cones, and 
not hyperboloids. The teeth are no longer straight, but 
follow helical curves traced on the conical surfaces. Such 




Fig. 219. 

wheels correspond in fact to screw-wheels, and have recently 
met with considerable favor on the continent of Europe, 
several French and German gear -cutting machines being 
specially designed for producing them. The teeth of spiral 
bevel-wheels usually follow a conical helix of constant pitch, 
the projection on the base of the cone being an Archimedean 
spiral. 

The velocity ratios of bevel-gear mechanisms can be 
determined by aid of the principles discussed in §§68 
and 69, and these gears may be arranged so as to corre- 
spond with the various compound or epicyclic trains de- 
scribed in a previous chapter; they may include annular 
wheels, and the general methods of determining their veloc- 
ity ratios are the same as those employed in the cases 



328 



KINEMATICS OF MACHINES. 



previously discussed. We shall take a few examples which 
will make this clear. 

In Fig. 220 is represented a well-known gear* which 
finds wide application. It consists of four bevel- wheels 
a, h, c, d, the frame or fixed link being e ; of these wheels a 
and b are of the same size, and c and d are also equal. The 
axes of the wheels intersect at right angles as shown. 




Fig. 220. 
If we imagine the frame e to be the fixed link, it is ob- 
vious that the following relations will hold : 





^ae = —^de= ^ed y 






<^be = —<^ce = ^ec' 




We have also 
and 


(Oa,_PN (Ode^^ 
U)te PM (Obe ' 




Hence 


<^ae = h^ad ^^^ ^ea = i<^da' 




Similarly, 


^ce = h^cb and (O,, = ^0)^^, 





* The so-called '< Differential " Bevel-gear. 



SPHERIC MOTION, 



329 



We thus see that if a be the fixed link, e will revolve 
arotind the axis MO with one half the angular velocity of 
d, and in the same sense. 

A compound reverted bevel-gear train is shown in Fig. 
221; in this particular case the axes do not intersect at 
right angles. In order to find the angular velocity ratio 
of a and c we have 

PN . 10,, QS 



CO 



(O 



ad 



bd 



and 



bd 



CO, 



thus 



PM ^,, 

PN.QS 



QR' 



Cx) 



ad 



co^, PM.QR 

TR.QS 
~ TV.QR' 




Fig. 221. 



If now we denote the angles TOS, QOS, and ROS by or, 
^, and respectively, it is evident that 

co^^ sin {d — a) sin /? 
co^^ ~sin {d—p) sin a 



330 



KINEMATICS OF MACHINES. 



Let n be the numerical value of this expression. If now 
we imagine that c makes one revolution, d being fixed, a 
will make n revolutions in the same sense. Let the whole 
mechanism receive one revolution in the negative sense 
about the axis OS, so that c is brought to rest, and d makes 
one revolution in the negative sense. While d makes one 
revolution the link a will now be making n—i revolutions 
in the sense opposite to that of the rotation of d. We then 
find that 



CO 



„. cot a — cot B 



co^^ cot ^ — cot ^* 

It should be noted that if the axes are at right angles, 



CO 



n = 



ad 



CO 



TR 

cd~'^QR 



cot « 
c^' 



and 



CO, 



CO 



=n— I 



dc 



sin (/?—«') 
sin (X. cos ^' 



Fig. 22 2 shows the way in which the bevel-gear train of 




Fig. 222. 



Fig. 2 20 is employed as a two-speed gear for transmitting 
power from the engine of an automobile to the driving- 
wheels. The engine-shaft a is rigidly connected to a bevel- 



SPHERIC MOTION. 331 

wheel a^. A second bevel-wheel, b, similar to a^ runs freely 
on the shaft, but can be held at pleasure by the application 
of a band-brake e to its brake-drum b^. The wheel &2 is 
rigidly connected to b and can be driven by the engine 
fly-wheel a^ by means of a friction- clutch c, the details of 
which are not shown. 

A pair of smaller bevel- wheels d^, d^ are carried by a 
frame /, which runs freely on the engine-shaft, and has upon 
it the sprocket-wheel /i, which gears with the pitch-chain 
that transmits the motion to the driving-wheels (see Fig. 
167). The fixed piece or frame of the mechanism is g. 

The gear can be worked under the following four condi- 
tions : 

(i) Engine running freely. The clutch c is discon- 
nected, / remains stationary, the brake e is not applied, 
and b rotates with angular velocity equal to that of the 
engine-shaft, but in the reverse sense. 

(2) Engine drives sprocket-wheel at low speed. In this 
case b^ is held by the brake e, the clutch c is not in gear, 
and / rotates in the same sense as a^, but with only half its 
angular velocity. 

(3) Engine drives sprocket-wheel at high speed. The 
clutch c now connects a^ and. b^, the brake e is not applied, 
a,b, and / therefore all revolve together with the same speed, 
and the bevel- wheels have no relative motion at all. 

(4) Carriage is stopped by applying the brake e, the 
clutch c being in gear, thus connecting b and a. 

99. Roller Bearings Involving Spheric Motion. — We have 
seen that the various links in bevel-gear mechanisms 
have relative spheric motion. Many roller and ball bear- 
ings, which may be regarded kinematically as augmented 
turning pairs, present examples of similar relative move- 
ment, which will next be discussed. The arrangement 
shown in Fig. 223 is often employed as a thrust bearing for 
a shaft or pivot on which a longitudinal pressure is exerted. 
The plates or flanges on which the pressure is taken are 



332 



KINEMATICS OF MACHINES. 



separated by a number of conical rollers, cages or frames 
being usually provided so as to prevent the rollers from 
getting out of position. In such a case, if we know cu^^ 
(the angular velocity of the flange or collar a with regard 
to the fixed link h), the angular velocity of the rollers can 
be exhibited in a very simple manner. In Fig. 223, with 
reference to the fixed link the roller c is turning about the 
instantaneous axis OB, while OA is its instantaneous axis with 




Fig. 223. 

relation to a. OX is the axis about which a is turning 
relatively to h. The real motion of c, relatively to 6, 
ma}^ therefore be considered as being the resultant of the 
motions of c relatively to a, and of a relatively to h. 

From a pole P we draw the vector PQ, representing co^^, 
according to the convention of § 16. The line PR is 
then drawn parallel to BO, the instantaneous axis of c 
with regard to h, and it meets a line QR drawn parallel to 
AO. Plainly c revolves relatively to 6 in a sense shown by 
the arrow-head on PR, and the arrow-head on RQ gives the 
sense in which a turns with regard to c. PQR is then a 
triangle of angular velocities in which 

and PQ, the resultant of QR and RP, must represent the 
resultant of oj^^ and a;^^, namely, w^j^. 



SPHERIC MOTION, 



333 



The motion of the roller c with respect to the links with 
which it pairs is evidently the same as that of a bevel- wheel, 
and the whole really forms a spheric mechanism. 

Next suppose that the rollers are cylindrical, as in Fig. 
224, and, as before, let OA, OB be the virtual axes of a and 
c and of c and h. The motion of the roller c relatively to 
a or to 6, may now be shown to consist of rolling about an 
axis parallel to OC, combined with a- spin about an axis 
parallel to AB. If we draw the triangle of velocities, as 
before, and proceed to resolve oj^j^ into two components, 
one {PS) parallel to the surfaces in contact, and the other 




Fig. 224. 

{SR) normal to those surfaces, we may evidently consider 
the real motion of c relatively to h as being due to a roll- 
ing PS combined with a spinning SR. The latter line then 
represents the angular velocity with which c is being twisted 
about a vertical axis in order to compel it to travel round 
in a circular path. In the same way RT gives the spinning 
velocity of a relatively to c. In Fig. 223 the form of the 
rollers is such that w^^ has no component normal to the sur- 
face of contact ; hence in that case there is, as has already 
been seen, no spinning, but a relative motion of rolling only. 
In Fig. 224 the line ST gives the total spinning at A and 
at B\ this is evidently equal to PQ, the relative angular 



334 KINEMATICS OF MACHINES. 

velocity of a and h about the axis OX. The condition for 
rolHng without spinning is that the instantaneous axis of 
the two bodies shall lie in the tangent plane at the point 
of contact (§ 6); this is the case in Fig. 223, but not in 
Fig. 224. 

Roller-bearings are often used for the journals of shafts ; 
but the rollers are then parallel, instead of conical, and 
their motion is not spheric but plane. 

100 Ball-bearings. — When balls are substituted for 
rollers, the motion is in general spheric, and Fig. 225 shows 
diagrammatically the arrangement of a number of typical 
ball-bearings, as applied to shaft- journals (a, h, c) and to a 
footstep- or thrust -bearing {d) . It will be seen that these 
bearings may be classified into two-, three-, or four-point 
bearings, according to the number of points of contact be- 
tween each ball and its races. The form (a) is a two-point 
bearing, and in practice it is necessary to give the working 
surfaces of the races a slight curvature in the direction of 
the axis of the shaft, as shown, in order to keep the balls 
in position. At {h) is shown a four-point bearing, and at {c) 
a two-point bearing, as used for the wheel or crank-axle 
bearing of a bicycle; {d) is a three-point thrust-bearing. 
Both the balls and their races are made of the hardest suit- 
able material, so as to reduce as far as possible the surface 
of contact. If the geometrical forms of the parts were 
exact, and if the material were perfectly rigid, contact 
would take place at points only, and any loss due to spinning 
friction at such points of contact would be negligible. 
Actually, however, this loss is quite appreciable, especially 
where the balls are heavily loaded, and, other things being 
equal, the best ball-bearing will be one in which this spinning 
of the balls about axes normal to their surfaces of contact 
is reduced to a minimum. The loss from rolling friction 
is not of such great importance. The method of deter- 
mining the rolling and spinning velocities of the balls will 
now be considered for various cases. 



SPHERIC MOTION. 



335 



In Fig. 226 is represented a chain consisting of a fixed 
cone h, and a ball c rotating in contact with h and also in 






id) 
Fig. 225. 
contact with the movable cone a. This movable cone 



336 



KINEMATICS OF MACHINES. 



rotates relatively to h about their common axis XY. It is 
required to determine the motion of the ball c. The whole 
arrangement may be supposed to form one portion of a two- 
point ball-bearing, the cones a and h being the tangent 
cones to the curved races in such a bearing as Fig. 225 {c). 

In order to see more clearly the nature of the relative 
movement of the ball and cones, suppose the ball to be 



h FIXED 




^ab 



Fig. 226. 



replaced by a very thin cylindrical roller of diameter d and 

At 
thickness J/, so that AA^=BB^ = — . Let co^^ be the rela- 
tive angular velocity of a and b ; then the linear velocity of 
the point A in a direction normal to the plane of the paper 
will be (o^f^xAM. The velocity of a point A^ (on the cone) 

/ Jt . \ . 
will be co^j^XiAM ^ sin^ I, while the velocity of the point 

A I (on the roller) will be the same as that of the point A^, 
namely, oj^f^XAM; that is, if we suppose the roller to be 
moving about the instantaneous axis BY, or if we suppose 
there is pure rolling at B. Hence A^ on the cone and A^ on 
the roller will have a relative linear velocity (normal to the 
plane of the paper) equal to 



SPHERIC MOTION, 337 

and their relative angular velocity about the axis AB will be 

M . 
-J^ = ^..sin^, 

2 

which is in fact the rate at which the ball will spin on the 
cone a if it rolls without spinning on the cone h. In other 
words, the total relative spinning of the ball with regard to 
the cones is ct>^^ sin cp. 

In general we have no right to assume that ^y is the 
virtual axis of c and 6, and that AY is the virtual axis of 
c and a. It is just as likely that these axes are BX and 
AX, in which case there would be a spin ^^^ sin ^ at 5 and 
pure rolling at A. We know that the centre of the ball 
travels in a circular path whose plane is perpendicular to 
XY, and that the three virtual axes lie in one plane. That 
the ball has spheric motion will be seen if we imagine that 
the whole bearing receives such a velocity that the centre of 
the ball remains at rest, while h and a rotate about XY with 
angular velocities of different senses. The points A and 
B are now moving perpendicularly to the plane of the paper, 
and the axis of rotation of the ball must therefore be some 
line lying in the plane of the paper, and cutting XF in some 
point 0. The position of will depend on the relative 
amount of rolling and spinning at A and B, and it forms the 
centre of the spheric motion for the whole bearing, so that 
the virtual axes of c and h and of c and a must pass through 
0. The line OC will of course be the axis of rotation of 
the ball, supposing the whole bearing to be moved so that 
C is at rest. To obtain the real motion of the ball when h 
is fixed, we must compound the rotation about OC with 
another about XY. It seems reasonable to suppose 
that if the surfaces of contact are equally rough, the total 
spin of the ball on the cones will be equally divided between 
A and B. The various virtual axes will then be as shown 



338 



KINEMATICS OF MACHINES. 



in Fig. 227, where ^0 is the virtual axis of a and c, and BO 
is the virtual axis of c and h. The point is found by draw- 
ing BO parallel to CX when C is the centre of the ball. The 
various velocities are easily found graphically, after the 
manner of Fig. 224. In Fig. 227 draw PQ representing co^j^ to 




Q 



do 





S 




Fig. 227 





any convenient scale; then PR and QR, drawn parallel 
respectively to AO and BO, will lepresent w^. and w^^. Each 
of these velocities can be resolved into a spin and a roll; 
thus PS is the angular velocity of rolling, and RS that of 
spinning, at A, while QT and TR are the rolling and spin- 
ning velocities at B. We have to show that RT = RS. 
Draw RU parallel to AX or BY, and produce BO to meet 
AX in Z. Then the triangles AOX, RPU are similar, and 
the triangles ZOX, RQU are similar, so that 

PU _PU_ UR_ 
UQ ~ UR ' UQ 



SPHERIC MOTION. 
OX ZX 



339 



~AX ' OX 
ZX 

~ AX' 

But by construction AX=ZX, since AC = CB. Therefore 
PU = UQ and SR=RT, so that the spin at A is equal to 
that at B. 

It is noteworthy that ST represents the total spinning 
at A and B, which is readily seen to be equal to co^^ sin ^. 




Fig. 228. 

The difference of the rolHng velocities at A and B is evi- 
dently CO^^ COS (f. 

In order to find the angular velocity of the centre of the 
ball around XY, we draw RV parallel to OC. We thus 
resolve QR, the real angular velocity of c about the axis OB 
relatively to the fixed Hnk, into QV, a rotation about XY, 
and VR, a rotation about OC. QV would then be the angu- 



340 KINEMATICS OF MACHINES. 

lar velocity to be given to the whole bearing so as to bring 
the centre of the balls to rest. 

The next example is a three-point thrust-bearing of the 
type of Fig. 225 (6/), as shown in Fig. 228. 

In this case it is evident that the ball must roll relatively 
to a about a virtual axis OA^A^, passing through both points 
of contact A^A^, while OB must be the virtual axis of c rela- 
tively to the fixed plate b. Let a and /? be the angles made 
by the virtual axes with the surfaces of contact, then the 
velocity diagram PQR is drawn as in the previous example. 
On resolving oj^^ and oj^^ into their component spinning and 
rolling velocities, as shown by the dotted lines, we find that 

Spinning velocity at A^=QT^ = (o^^ sin a ; 

B =^RS =0;^, sin/? 

= (0. 



< I 



ab' 



We find also that 



Rolling velocity at A^=RT^ = a)^^ cos a ; 
'' B= PS = aj^, COS l^ 



The relative angular velocities for a three-point bearing 
in the more usual case where OA^A^ is not perpendicular 
to OX will be determined in a numerical example, after 
considering the case of a four-point bearing, which may be 
worked out by similar methods. 

In Fig. 229 A^A^ ^1^2 ^^^ "t^^ points of contact, and 
plainly if there is to be spheric motion on the part of 
c, A^A^ and B^B^ must meet at some point on XY. 
If this is not the case, slipping will occur at some one 
or more of the points of contact, and the relative motion 
at these points will no longer consist simply of rolling 
and spinning combined. In the figure OA^A^ and OB^B^ 
are the virtual axes of a and c, and of h and c, respectively. 
Let a; J) be the angular velocity of the centre of the ball, 



SPHERIC MOTION. 



341 



SO that the linear velocity of the point C in a direc- 
tion normal to the plane of the paper will be w^xCL. 
Since A is a point on OA^A^, its linear velocity will be 
ctj^f^XAM in the same direction. But b is the fixed link 



b (fixed) 




'Wa6 



R -(Waft- Wo) 




-^ ^«• 



Q Wad P 

Fig. 229. 

and c is rolling upon it about the axis OB, so that we may 
also say that the linear velocity of the point C is o^^^ X CB, 
and that of A is co^j^xAN. Hence, 

, co^XCL = aj^^XCB, 
w^^XAM = (o^^XAN. 



Thus 



0)^ CB.AM 



oj^, CL.AN' 



342 KINEMATICS OF MACHINES. 

The angular velocity of the centre of ball relatively to a 
is readily found if we imagine that the whole bearing has 
given to it an angular velocity — co^^. The centre of ball 
will then be moving with an angular velocity {oj^^ — oj^) 
about the axis XF in the same sense as oj^^, and a will be at 
rest. Then 




^-CA\^ c.,,r^^' 



(o^^ CL CL.AN-CB.AM 



(o^, CA' CL.AN 

_ CL.AN-CB.AM 
CA.AN 

In the same way we may find an expression for oj^^; 
for oj^^xAN = (o^^ X AM, so that 

oj^, AM 



ao 



The values of the various angular velocities may also be 
obtained graphically, as was done in the case of the three- 
point bearmg of Fig. 228. Draw the triangle PQR (Fig. 
229) representing (o^^^, oj^^, and co^^. Then we can resolve a>^^ 
into cjq, the angular velocity of the centre of the ball 
around XY, and (o^, the angular velocity about OC, 
Similarly oj^^ can be regarded as the resultant of a veloc- 
ity (x)^ about OC and a velocity —(^0^,-^0) about XF. 

In order to find the spinning and rolling at the points of 
contact B^ and B.^, co^^ is to be resolved along and perpen- 
dicular to the surfaces of contact, giving 

angular velocity of spinning at 5j =QTi ; 
''rolling at B,=RT,; 
" " spinning at ^2=7^72; 
" rolling at B, = QT,. 

In the same way the velocities at A^ and A^ may be found. 



SPHERIC MOTION, 



343 



Taking a numerical example, we may determine the 
various velocities in the three-point bearing shown in Fig. 
230, the dimensions being: 



Diameter of balls 0.25 inch 

Distance of ball centre from axis (CL) ..0.5 ** 

Angle of cone of ball-race h 30° 

** a 30° 

'' ^ • 



20^ 




100 
I 



400 REVS. PER WIN. 



Fig. 230. 



344 KINEMATICS OF MACHINES. 

After drawing CB, CL, AN, AM, we find CA=o.ioS, 
CE =0.115, CL = 0.400, AN = o.2iS, AM =0.309. Let us 
suppose that co^^ is 20c revolutions per minute. Then the 
velocity of centre of balls 

_ C^ ■ AM _ 200X0. 1 15X0.309 

"^o-^abX CL.AN 0.400X0.218 

= 81.2 r.p.m. about axis XY, 

(CL CB.AM ) 
Again, a.^^==o.^^xl^--^j-j^l 

0.400 0.115X0.309) 



200 



and 



0.108 0.108X0.2 
= 440 r.p.m. about axis OA, 



0.300 

0). = 200 X 5- =284 r.p.m. about OB, 

"^ 0.218 ^ ^ 



In order to determine the spinning and rolling we have 

Angular velocity of spinning at B^ = w^^ sin /9 
(SR on diagram) = 284 X 0.342 

= 100.5 r.p.m. 
Angular velocity of rolling at B^ =^cb ^^s /? 
{SP on diagram) = 284 X 0.9396 

= 267 r.p.m. 
Angular velocity of spinning at A^or A^ = co^^ sin a 
(QT 2 or QT^ on diagram) = 440 X o . 5 

= 220 r.p.m. 
Angular velocity of rolling at A 2 or A ^ =^ac ^^s a 
{R T^ or R Ti on diagram) ^440X0.866 

= 381 r.p.m. 

The reader should check these numbers by drawing the 
velocity diagram for himself, and measuring the various 
lines representing the velocities. 

To compare the relative merits of different ball-bearings 
we should have to calculate not only the velocities, but also 



SPHERIC MOTION, 345 

the pressures between the surfaces at the points of contact ; 
the work wasted in various cases could then be estimated. 
This part of the work, however, belongs to Dynamics 
rather than to Kinematics. 

For further information, on the subject of ball-bearings 
the reader is referred to Sharp's "Bicycles and Tricycles," 
Chapter XXV, and to a number of papers in various engi- 
neering periodicals.* 

* Engineerings April 12, 1901. Zeitschrift d. V. D, /., Jan. 27, 1900; ibid.y 
Jan., 1901, pp. 73 and 119; ibid.^ Jan., 1901, p. 332. 



CHAPTER XIII. 

KINEMATIC CLASSIFICATION OF MECHANISMS. 

10 1. Historical Sketch'. — In treating of the theory of 
Mechanisms, it has been the aim of many writers to devise 
some method of analysis whereby mechanical contrivances 
in general might be resolved into their several component 
parts, capable of being represented, if necessary, by symbols, 
and capable also of being recombined in such a fashion as to 
produce new mechanisms. Such a system, if complete and 
workable, would of course be of great service to the inven- 
tor, and would save him from the fate, only too common, 
of designing with great toil some device which has been 
known and used for years. In the words of Willis, ' ' there 
appears no reason why the construction of a machine for 
a given purpose should not, like any usual problem, be so 
reduced to the dominion of the mathematician as to enable 
him to obtain, by direct and certain methods, all the forms 
and arrangements that are applicable to the desired pur- 
pose, from which he may select at pleasure." It must be 
confessed that so far no such system of analysis and synthe- 
sis has been found of any great practical value; many of 
the proposals, however, are interesting and suggestive, and 
a brief account of some of them will not be out of place in 
this book. Before entering upon it we may glance at the 
historical development of the subject of the Kinematics 
of Machines. 

A book dating from the eighteenth century* seems to 
be the first treatise on machines which can be considered at 

* Leupold. Theatrum Machinarum. 1724. 

346 



KINEMATIC CLASSIFICATION OF MECHANISMS. 347 

all systematic. Leupold's predecessors had indeed de- 
scribed sundry machines and devices, but their order of 
arrangement was always arbitrary,- and no attempt was 
made to study machines by considering the relative motions 
of their parts. The theory of machines, treated either from 
the kinematic or dynamic standpoint, did not in fact exist. 

Euler * taught that the motions of rigid bodies should be 
investigated by the methods of geometry, as well as by the 
aid of dynamics, but it does not appear that he had in view 
the special application of these principles to the motions of 
the parts of machines. Monge in 1794 conceived the idea 
of treating machines as contrivances for changing one kind 
of motion into another, and was the first to suggest that the 
essential "elements of machines" should be enumerated 
and studied. His system formed the basis of the course 
adopted in the Ecole Polytechnique soon after its founda- 
tion — a course laid out by Lanz and Betancourt,t and classi- 
fying the m.otions of the parts of machines as (i) rectilinear, 
(2) circular, or (3) curvilinear. Combinations of these 
motions are considered, while each motion may be contin- 
uous or alternate. The work of Lanz and Betancourt 
was incomplete, because no attempt was made to calculate 
these various motions ; their scheme underwent many mod- 
ifications, and has not survived. A system somewhat 
similar in intention, but differing in detail, was propounded 
by Borgnis.t It has met with the same fate. 

It IS to the physicist Ampere § that we owe an important 
advance. He saw clearly that a mechanism should be 
studied as " an instrument by the help of which the direction 
and velocity of a given motion can be altered" ; thus going 
further than Euler, and laying the foundations of that science 
of Machines to which, in accordance with his suggestion, we 
apply the name Kinematics. 

* Euler. Theoria Motus Corporum. 1765. 

■j" Lanz and Betancourt- Essai sur la composition des Machines. 1808. 
J Borgnis. Traite complet de Mecanique appliquee aux Arts. 1S18. 
§ Ampere. Essai sur la philosophie des Sciences. 1834 



348 KINEMATICS OF MACHINES. 

Ampere was followed by Willis,* who confined himself 
to the consideration of what he termed the * ' Elements of 
Pure Mechanisms," and did not deal with the ' ' generalities of 
motion." The ** Principles of Mechanism" takes a less 
abstract view of the science of Kinematics than Ampere 
seems to have held, and in that book the author endeavors 
to form a system embracing all the elementary combina- 
tions of mechanism, and admitting of an investigation of 
their, modifications of motion. He does not attempt to 
deal w4th dynamical questions, but gives practical and use- 
ful solutions of many leading problems in applied kinematics. 
His system of classification will receive some consideration 
in a later section; we shall see that its groundwork is the 
mode in which the motion is transmitted, or, as we should 
now express it, the kind of relative motion existing, in. 
various mechanisms. 

In several of his books Rankine t deals with kinematical 
questions, treated under such titles as the Geometry of 
Machinery and the Theory of Mechanism. His views were in 
some few respects erroneous and incomplete, and his nomen- 
clature has not been followed to any large extent, but his 
system of dealing with the motion of machine parts by the 
aid of instantaneous centres, and his methods of solving 
certain special problems, were in many cases far more power- 
ful and effective than any previously employed. 

The appearance in collected form of the kinematical 
writings of Reuleaux X furnished students with the first 
text -book whose methods have met with really wide accept- 
ance. It is to Reuleaux that we owe the idea of a mechan- 
ism regarded as a chain made up of links any one of which 
may be considered as being fixed. Starting with this con- 

* Willis Principles of Mechanism. 1841. (Second Edition 1870.) 
f Rankine. Applied Mechanics. 1858. 

Manual of Machinery and Miilwork. 1869 
J Reuleaux. Theoretische Kinematik. English Translation by Dr. Kennedy- 
1876. 



KINEMATIC CLASSIFICATION OF MECHANISMS. 349 

ception, and taking account of the relative motion of these 
links as determined by the pairing of their elements, we are 
led to a wide and comprehensive view of the whole kinematic 
theory of mechanisms. The earlier work of Reuleaux has 
now been supplemented by the publication of a second part 
of his text-book.* 

Burmester's important treatise t is not so well known to 
English-speaking readers as it should be. Only the first 
volume, dealing with plane motion, has yet been published. 
Burmester's method of treatment differs from that of Reu- 
leaux in making a more liberal use of purely mathematical 
and geometrical principles, but the two authors agree in 
their fundamental conception of the subject, and, to a large 
extent, in their nomenclature and definitions. A consider- 
able amount of space is devoted by Burmester to the kine- 
matics of a plane rigid system ; he deals with the principles 
of constraint in plane motion, and passes on to the consider- 
ation of plane mechanisms and the relative displacement, 
velocity, and acceleration of their various parts. The second 
volume is to treat, after a similar fashion, of non-plane 
motion. 

102. Classification of Willis. Babbage's Notation. — The 
following sections contain a short account of some of the 
schemes suggested for classifying and symbolizing the vari- 
ous kinds of mechanisms. 

Like almost all his predecessors, Willis contented him- 
self with proposing a scheme of classification without en- 
deavoring to invent any notation, or system of signs, by 
which a given mechanism could be represented by a formula. 
Without apparent reason, Willis excludes from his system 
all hydraulic machines. Some other classes of mechanism, 
for example those including springs, are also omitted. In 
fact he considers as "pure mechanisms" only certain types 



* Reuleaux. Die praktischen Beziehungen der Kinematik zu Geometric und 

Mechanik. 1900. 
I Burmester. Lehrbuch der Kinematik. i\ 



350 KINEMATICS OF MACHINES. 

of machines, which seem to have been selected in a some- 
what arbitrary fashion. In these machines, according to 
WilHs, motion is transmitted in * * elementary combina- 
tions" by five methods, namely: 

Division. Method of Transmission. Example. 

A. By rolling contact. Toothed gearing of various 

sorts. 

B. By sliding contact. Cams, screws, worm- and 

screw-gearing, escape- 
ments. 

C. By wrapping connection. Bands, chains, and other 

gearing. 

D. By linkwork. Cranks, eccentrics, and 

other linkwork. Ratchet- 
wheels and clicks. 

E. By reduplication. Tackle of all sorts. 

Each of these five main divisions is again separated into 
three classes, in which the velocity ratio is either (a) con- 
stant, {h) varying, and {c) constant or varying; while due 
regard is had to the question whether the ' ' directional rela- 
tion" is constant of varying. 

This system or classification has not been widely used, 
and possesses certain manifest imperfections. It was, how- 
ever, a great advance on that of Lanz and Betancourt or 
on that of Borgnis, because it was designed with a view of 
facilitating calculations regarding the relative motions, or 
velocity ratios, in mechanisms, rather than with the aim 
of classifying mechanisms for purely descriptive purposes. 

In the ''Principles of Mechanism" Willis devotes some 
space to the exposition of the scheme of notation proposed 
by Babbage ; * a scheme devised by that ingenious inventor 
primarily for the purpose of clearly representing the rela- 
tions of the parts of his calculating-machine, and especially 

* A Method of Expressing by Signs the Action of Machinery. Phil. Trans., 
1826. 



KINEMATIC CLASSIFICATION OF MECHANISMS. 351 

applicable to complex trains of wheel and ratchet gearing. 
As this notation involves the construction of an elaborate 
sheet or diagram for each machine, it by no means answers 
the purpose of a system such as that of Reuleaux, which will 
be described later, where each mechanism is to be denoted 
by a formula of three or more symbols. Babbage's method 
of notation corresponds more closely to that employed by 
clock and watch-makers, in which the various wheels are 
represented by the numbers of their teeth, written in suc- 
cessive lines, placing vertically over each other the numbers 
of wheels which gear together. Thus 

48 
6-45 

6-30 

would represent a wheel-train comprising a ''great wheel'* 
of 48 teeth gearing with a pinion of 6 teeth, the pinion-arbor 
or axis carrying a second wheel of 45 teeth, gearing in its 
turn with a 6 -tooth pinion whose arbor carries an escape- 
wheel of 30 teeth. Babbage, however, shows on his dia- 
gram the kind of motion, whether uniform, intermittent, 
variable, or continuous, of each part with relation to the 
frame of the machine, and Willis gives an interesting exam- 
ple* of such a diagram, as constructed for a sawmill. It 
would appear that Babbage's notation, while extremely 
convenient in certain cases, by no means answers the pur- 
pose of a general scheme by means of which the mode 
of action and relative motions in any given mechanism 
may be indicated. 

103. Classification and Notation of Reuleaux. — Such a 
system has been devised by Reuleaux, f and is explained and 
used in his text-book. It is intended to be perfectly general 
in its application, and includes signs of three kinds, which 
denote (i) the class or name of the body or link referred 

* Principles of Mechanism, Ed. 1870, p. 288. 
\ Kinematics of Machinery, English Ed., p. 251. 



352 KINEMATICS OF MACHINES. 

to, as distinguished by its geometrical shape or its nature ; 
(2) the form of the body, whether soHd or full, or hollow or 
open, whether plane or curved; and lastly, (3) the relation 
of one element to its companion, or of one liiik to the next 
in the chain. Some special symbols are required to indicate 
incomplete pairs, methods of closure, and so on. 

In the first division the following name symbols have been 
adopted : 



s 


Screw. 


G 


Sphere or globe. 


R 


Solid of revolution. 


A 


Sector or arc. 


P 


Prism. 


Z 


Tooth or projection. 


C 


Cylinder. 


V 


Vessel or chamber. 


K 


Cone. 


T 


Tension-organ. 


H 


Hyperboloid. 


Q 


Pressure-organ. 



These symbols require no explanation. 

With regard to the next kind of symbols, those of form^ 
it is evidently necessary to indicate among other particulars 
whether a given body is full, open, or plane; whether its 
profile is curved or non-circular, or has upon it teeth; or 
whether its profile or section is prismatic. A link, as we 
have seen, may be liquid or gaseous, and a large number of 
other cases may be suggested, all of which should be covered 
by any general system of symbols. Reuleaux proposes to 
do this by adding to the Roman capital letters which he 
selects as the name-symbols, certain form-symbols, written 
to the right of the name-symbol, and either above or be- 
low it. A few examples will illustrate the way of doing 
this. We may use the following : 

■Y full or solid. ° plane. 

— open "or hollow. '^ curved profile. 

From these and the preceding symbols we have, among 
many others : 

C+ full cylinder. C~ open cylinder. 

S+ screw or bolt. . S~ nut. 



KINEMATIC CLASSIFICATION OF MECHANISMS. 353 

P— hollow or open prism whose base has a curved outline. 

Cf non-circular spur-wheel with external teeth. 

K° face-wheel (plane bevel- wheel) . 

Tp prismatic tension-organ (flat belt). 

T^ circular tension-organ (wire). 

Qx liquid pressure-organ. 

Qy gas, air, steam, etc. 

V~ cylinder of an engine or pump. 

V+ the piston working in it. 

The third class of symbols express relation, as regards 
pairing, hnkage, or fixing, or as regards position and mag- 
nitude. Pairing is denoted by a comma, linkage by a dot 
or dotted line ; a fixed link is indicated by underhning, and 
the usual signs are employed for equality, parallelism, and so 
on. For example: 

C~ . . . C~ link connecting two open cylinders or eyes. 
P^, Ct rack pairing with a pinion. 

S+ ^ S~ screw pair, screw and nut being of course equal 

in size. 

Incompleteness is indicated by the use of the sign of 
division, so that we get : 



2 

7 



portion of an open cylinder. 

a full cylinder paired by force-closure. 



The method of closure is indicated by the divisor. 

As an example of the method of writing out the formula 
for a simple mechanism, we may refer to Fig. 133. The 
spur-wheel mechanism acd would be written {d being the 
fixed Imk) : 

C+ . . . I . . . C + , C+. . . I . . . C+ c- .. . II ...c- 



(referringto link a, link 6, and link J). 



354 KINEMATICS OF MACHINES. 

Here | means con-axial ; || means parallel. 

After describing and enumerating the various symbols, 
Reuleaux proceeds to show how the resulting formulae may- 
be shortened. He employs (S), (C), and (P) for a screw 
pair, a turning pair, and a sliding pair, respectively, and 

_^ 
would write (CJ')« for "a chain formed of four links, each 

connecting two parallel cylindric elements"; d being the 

fixed link and a the one which drives the chain. This is of 

course the quadric cylindric crank chain. His symbols 

have a very wide range of adaptability ; the reader will be 

interested, for example, in the formula for a paddle-steamer, 

which is 

C+ ... I ... C+, Q, ... Q„ V- ... + ... Cz 

This may be contracted to (C' C^^Vx) « . 

Here h is the liquid link, a the paddle-wheels, and c the 
ship itself. Vx is a contraction for V~, Q^, and C^x for 
C^, Qa; + is the sign for ''crossed at right angles" when 
used as the symbol of relation of the elements of a link. 

The original text-book must of course be consulted if 
any real acquaintance with the scheme is desired ; the exam- 
ples given here will serve to indicate the scope and posi- 
bilities of the system. 

104. Classification of Hearson. — The most recent system 
of notation devised by Professor Hearson * differs essentially 
from that of Reuleaux, for it is based on a somewhat different 
conception of the meaning of the term machine. Hearson 
considers that a machine is to be regarded as "an embodi- 
ment of a combination of elementary motions (of which it 
will be found that the number of kinds is comparatively 
limited)"; these "elementary motions" being the relative 
motions of the machine-parts. He treats first of plane 
mechanisms, and suggests the following symbols : 

* Phil. Trans., 1896, Vol 187, p. 15. 



KINEMATIC CLASSIFICATION OF MECHANISMS. 355 

a motion of complete and continuous relative rotation. 
U a swinging to-and-fro motion, like that of a pendulum, 

consisting of successive partial rotations in opposite 
senses. 

1 a sliding motion. 

Taking a four-link mechanism as the general case (three 
and five or more links being inadmissible in simple machines 
for reasons already given), it is shown that there may be in 
such a mechanism either 

four O motions, 
or three O's and one U \ under certain conditions as 

or two O's and two U's, ) to length of links, 

or four U motions; 

while it is impossible to have one O motion only and three U's. 

On considering the substitution of I motions for O's and 
U's, it is found that (in all) fourteen combinations of O's, 
U's, and I's are permissible. 

In order to denote such motions as that of the teeth of a 
pair of spur-wheels, Hearson assigns the symbols 

W for a combination of two U motions ; 
C30 for a combination of two O motions ; 
C for the wrapping motion of a belt on its pulley. 

He further proposes to distinguish betw^een the sense of 
motions by the use of large and small letters, so that, for 
example, tw^o pulleys mounted on a frame joined by a crossed 
belt would be OcCO or oCco ; if an open belt were employed 
the formula would become OcCo or oCcO. 

Passing to spherical mechanisms, a srmilar system is 
outlined; certain limitations, however, are imposed by 
the differences between plane and spherical geometrical 
relations. 

Adopting the symbol V for helical motion having a con- 
stant pitch ratio, and H for one in which the pitch ratio 
varies, it is found that we may have the mechanisms UVl, 
VVl, VVU, and VVV, of which there are eight inversions in 



35^ KINEMATICS OF MACHINES, 

all. With H motions four combinations, with eleven inver- 
sions, give us UHI, VHI, VHU, and VHV; these may be 
classed under the head of cylindrical mechanisms. Hearson 
proposes to group all remaining simple mechanisms in a 
fourth division, comprising those in which the axes neither 
meet nor are parallel. He then discusses compound mech- 
anisms in which there are more than four separately moving 
pieces, and yet the motions are of a determinate character. 
This leads to a method of formulating such mechanisms ; it 
will be sufficient here to give as an example the formula for 
the portion of a locomotive consisting of frame, piston and 
rod, connecting-rod, crank-axle, coupling-rod, and the crank 
of a second coupled driving-wheel. This is 



6-o-I-U-d-O. 
I I 



To explain this formula, it is to be noted that the link 
shown by the thick line is the frame or the link which is re- 
garded as being fixed. The links which move in contact with 
it are the piston and rod (U — I) , the crank-axle (o — O) , and 
the coupled driving-wheel and its axle (O — o) . The connect- 
ing-rod will be denoted by (U — o), and the coupling-rod by 
(O — o) , the frame being (o — I — O) . Here the large letters 
refer to turning motions in which the angle is increasing, 
while small letters indicate those in which the angle is dimin- 
ishing. The lines connecting the letters of the complete 
formula show which motions are possessed by the various 
links. Hearson' s scheme does not appear to contemplate 
the inclusion of fluid links, and, as outlined in the paper 
cited, is by no means so complete as that of Reuleaux. 

105. Remarks on Classification. — It is not surprising that, 
up to the present, no system of kinematic classification has 
proved so simple, and at the same time so wide in its scope, 
as to be generally accepted as an assistance both to the in- 
ventor and to the student of the theory of mechanisms. 
The nomenclature and classification of Reuleaux and the 



KINEMATIC CLASSIFICATION OF MECHANISMS. 357 

suggestions of his critics * have rather an academic than a 
technological interest, and indeed it seems probable that the 
inevitable complexity of any such scheme, when it finally 
develops itself, will render it more suitable to the lecture- 
room than to the drawing-ofhce. 

From the instances already quoted it will be seen that 
kinematicians have taken two distinct points of view in 
regarding the nature of a machine, and this fundamental 
divergence has necessarily affected their ideas when search- 
ing for a scheme of classification. On the one hand we have 
the school followed by Hearson and others, who take (with 
considerable modification) the ideas of Monge and Willis as 
a basis, and look first at the relative motion of the machine- 
parts; on the other we have the school of Reuleaux, who 
consider first the forms of the elements, or working sur- 
faces, which control these relative motions. Whichever line 
is taken, we are soon driven to the conclusion that in many 
raechanisms the actual relative motions produced depend 
not only on the forms of the elements, but also on the way 
in which the driving forces and the resistances act on the 
various links of the chain. Reuleaux is thus led to his idea 
of force-closure of pairs and chains, which has perhaps met 
with more criticism than any other part of his work, and, 
to complete his classification, he is obliged to introduce 
indirectly certain dynamical conditions. 

While it is the province of the science of Kinematics of 
Machines to deal solely with questions of motion, apart 
from dynamic considerations, it does not seem probable that 
any really effective system of machine classification can be 
based simply on kinematic relations. 

* For example see Hearson, loc. cit.; Koenigs, Comptes Rendus^ IQOI? Aug. 15 
and 19, Sept. 2 and 23. 



INDEX. 



PAGE 

Acceleration 7 31. 33» I4i» i45 

Acceleration of Cable Car 53 

Acceleration of Connecting-rod no 

Acceleration of Cross-head ' 102 

Acceleration Curves 51 

Acceleration, Diagrams of 44, 49, 51 

Acceleration Images 155, 157 

Acceleration of Piston 102 

Acceleration, Polar Diagrams of 53i i55 

Acceleration, Radial 34 

Acceleration, Resultant 37 

Acceleration in Simple Harmonic Motion 61 

Acceleration, Uniform ; 31, 49, 217 

Accumulator, Hydraulic 269 

Addition, Vector 36 

Adjustable Escapements 237 

Adjustable Fluid Escapements 274 

Air-compressor 259 

Air-pump, Edwards' 269 

Alteration of Mechanisms 164 

Ampere 347 

Amplitude of Simple Harmonic Motion 58 

Anchor Escapement 238 

Anemometer ^ 283 

Angle, Pitch 278 

Angular Velocity 27, 32 

Angular Velocity in Ball-bearings 336 

Angular Velocities, Composition of 37 

Angular Velocity of Connecting-rod no 

Angular Velocity of CyHnder in Oscillating Engine 112 

Angular Velocity in Hooke's Joint 316 

Angular Velocities in Quadric Crank-chain 73 

Angular Velocity in Universal Joint 316 

Annular Whee 203, 209 

359 



360 INDEX. 

PAGE 

Anti-parallel Cranks 175 

Archimedean Drill 279 

Archimedean Spiral , 57, 216, 327 

Arc-lamps, Electric 230 

Atkinson's "Cycle" Gas-engine 160 

Augmentation of Chains 166 

Augmentation of Mechanisms 166 

Automobile 330 

Auxiliary Circle 58 

Average Velocity 29 

Axial Pitch 288 

Axis, Twist 279, 294 

Axis, Virtual , 12, 305 

Axle-box 171 

A.xode 12, 191, 305 

Axode, Twist 294 

Babbage's Notation 349 

Back Gear of Lathe 204 

Balance, Roberval 86 

Ball-bearings 167, 334 

Ball-bearings, Angular Velocity in 336 

Ball-bearings, Relative Spinning in 337 

Ball-and-socket Joint 21 

Base Circle 196 

Bearings, Ball 167, 334 

Bearings, Roller 331 

Bearing, Thrust 33i, 334 

Beam-engine 75, 141 

Bell-crank Levers 220 

Belts 24, 244 

Belt, Crossed 245 

Belt-gearing between Non-parallel Axes 254 

Belt-gearing, Variable Velocity Ratio in 246 

Belt-gearing, Velocity Ratio in 244 

Belt, Length of 246 

Belt, Open 245 

Belt-shifter 220 

Belt Transmission , 255 

Belt, Thickness of 251 

Betancourt ; 347 

Bevel-gear 317 

Bevel-gear, Compound Reverted 329 

Bevel-gear Cutting-machines 323 

Bevel-gear, Differential 328 

Bevel-gear, Epicyclic 328 



INDEX. 361 

PAGE 

Bevel-gear, Velocity Ratio in 329 

Bevel-wheels 306, 318 

Bevel-wheels, Internal 319 

Bevel- wheels, Involute Teeth in 321 

Bevel-wheels, Spiral 327 

Bevel-wheels, Teeth of 320 

Bevel-wheel Teeth, Setting out 322 

Bevel-wheels, Velocity Ratio in 319 

Bicycle, Front-driving, Gear of 208 

Bicycle, Free-wheel 228 

Bilgram Gear-cutting Machine 323 

Blade, Guide 283 

Block, Breech 282, 284 

Blower, Root 267 

Body, Restraining 181 

Borgnis 347 

Brake 232 

Brake, Friction 232 

Brake, Froude 271 

Brake-block 235 

Brake, Strap 272 

Breech-block 282, 284 

Bremme's Valve-gear 143, 151 

Bricard Straight-line Motion 94 

Brotherhood Steam-engine 173 

Buffer-spring 259 

Burmester, Professor 183, 349 

Cable-car. 53 

Calibre 282 

Cam, Cycloidal 226 

Cam, Cylindrical 215, 219 

Cam, Globoidal 226 

Cam, Involute 219 

Cam-pair 213 

Cam, Positive-motion ', 222, 226 

Cam, Rotating 215 

Cam, Sliding 215, 219 

Cam-train 213 

Cam-trains, Velocity Ratio in 222 

Capstan, Driving-gear of 211 

Cartwright's Straight-Une Motion 212 

Cell, Peaucellier 90 

Centre, Permanent 189 

Centre, Virtual 12 

Centrifugal Pump , 264, 285 



362 INDEX. 

PAGE 

Centrodes 12, 98, 170, 187 

Centrodes, Circular 190 

Centrodes in Quadric Crank-chain 72 

Centrodes, Reduced 170 

Centrodes of Slider-crank Chain 97 

Centrodes, Transformed 1 70 

Chains 6, 24, 244 

Chain, Closed 7 

Chain-closiure 140, 172, 173 

Chain-closure of Pairs 175 

Chain, Compound 7 

Chain, Conic 309 

Chain containing Sliding Pairs only 138 

Chain, Cylindric Crank 97 

Chain-gearing, Velocity Ratio in 251 

Chain, Inversion of 9 

Chain, Kinematic 6 

Chain, Pitch 253, 331 

Chain, Renold 254 

Chain, Simple 7 

Chain, Slider-crank 97 

Chains, Augmentation of 166 

Chains, Crossed-slide 168 

Chains, Incomplete 172 

Chains, Reduction of 168 

Chamber Crank-train 261 

Chamber Wheel-trains 264 

Chamber Wheel-trains, Epicychc 267 

Change-point 80, 1 72, 176 

Checking-ratchet 230, 271 

Chronometer 253 

Chuck, Self -centring 285 

Circle, Auxihary 58 

Circle, Base 196 

Circle, Gorge 300 

Circle, Pitch 193 

Circular Centrodes 190 

Circumferential Pitch 288 

Clamp, Kinematic 1 80 

Classification (Hearson's) 354 

Classification of Mechanisms 25, 346 

Classification (Reuleaux) 351 

Click 277 

Closed Chain 7 

Closed Pair 5 

Closure 164 



INDEX. 363 

PAG3 

Closure, Chain 140 

Closure, Force 140, 171, 173, 221 

Clutch 235 

Clutch, Coil 272 

Cochrane Engine 263 

Coil Clutch 272 

Common Pump 268 

Common Tangent 288 

Compensating Cylinders 132 

Composition of Angular Velocities 37 

Composition of Simple Harmonic Motions 67 

Composition of Velocities 35 

Compound Chain 7 

Compound Epicyclic Reverted Train 207 

Compound Reverted Bevel-gear 329 

Compound Wheel-trains 201 

Compressed-air Transmission 261 

Compressor, Air 259 

Conchoid 137 

Cone Pulleys 246 

Conic Chain 309 

Conic Mechanisms 309 

Conic Quadric Crank-chain 308 

Conical Rollers 331 

Conical Screw 284 

Connecting-rod, Acceleration of . . . . .' no 

Connecting-rod, Angular Velocity of no 

Connecting-rod of Locomotive 82 

Connecting-rod, Obliquity of 100 

Connecting-rod, Triangular 175 

Constant Acceleration 49 

Constrained Motion 3 

Constraint - 19 

Constraint, Degrees of 177 

Copying-press 279 

Corliss Engine 233 

Corliss Valve-gear 233 

Coupling, Oldham's 127 

Coupling-rod of Locomotive 82 

Crank 71 

Cranks, Anti-parallel 175 

Crank-chain, Conic Quadric 308 

Crank-chain, Double 77 

Crank-chain, Lever 77 

Crank-chain, Quadric 70 

Crank-pin, Velocity of 102 



3^4 INDEX. 

PAGK 

Cranks, Parallel 8i 

Crossed Belt 245 

Crossed Screw-chain.. 281 

Crossed-slide Chain 131, 168 

Crossed Slider-crank Chain 122 

Crossed Swinging-hlock Slider-crank 123, 

Crossed Swinging Slider-crank 123 

Crossed Turning-block Slider-crank 123 

Crossed Turning SHder-crank 123 

Crosshead, Acceleration of 99 

Crosshead, Displacement of 99 

Crosshead, Velocity of . 99 

Curve, Sine 66 

Curve Triangle, Equilateral 185 

Curve, Trochoidal 186 

Curves, Acceleration ; 51 

Curves, Cycloidal 195, 

Cutting-machines, Bevel-gear 323 

" Cycle " Gas-engine, Atkinson's 160- 

Cycloidal Cam 226 

Cycloidal Curves 195 

Cycloidal Wheel-teeth 197 

Cyhnder, Angular Velocity of, in Oscillating Engine 112 

Cylinders, . Compensating 13^ 

Cylindric Crank-chain 97 

Cylindrical Cam 215, 219 



Dashpot. 235 

Dead-centre 142^ 

Dead-point 80, 172^ 

Degrees of Constraint 177 

Degrees of Freedom 19, 177 

Describing Circle .... 197 

Design, Machine 9 

Diagrams of Acceleration — . .44, 49, 51 

Diagram of Acceleration of Street-car 46 

Diagrams of Displacement 38 

Diagram of Displacement for Train 43 

Diagrams, Polar Acceleration 155, i6a 

Diagrams, Polar 146 

Diagrams, Polar, Velocity 160 

Diagrams of Velocity 38, 49 

Diagram of Velocity of Street-car 46 

Diagram of Velocity for Train 43 

Difference, Phase 64 

Differential Bevel-gear 32& 



INDEX, 365 

PAGE 

Differential Pulley-block 252 

Differential Pump 269 

Direct-acting Engine 99, 1 59 

Direct-acting Steam-engine 99, 1 59, 261 

Direction 28, 156 

Displacement, Diagrams of 38 

Displacement, Polar Diagrams of ; 53 

Donkey-pump : 121 

Double-acting Pump 269 

Double-adjustment Plummer-block 140 

Double Crank ' 77 

Double Helical Wheels 296 

Double Lever 80 

Double Slider-crank Chain 123, 127, 263 

Drill. Archimedean 279 

Driving gear of Capstan 211 

Duangle 183 

Duplex Steam -pumps 132, 168 

Duplication of Mechanism 172. 173 

Eccentric ; 145. 153, ^64 

Eccentric-pin of Paddle-wheel 79 

Ecoie Polytechnique 347 

Edwards' Air-pump 269 

Electric Arc-lamps , 230 

Element, Kinematic . 3 

Elements, Expansion of , 164 

Elements. Forms of 186 

Ellipse 1 26, T 36. 295 

Elliptic Chuck , t 27, 1 30 

Elhptic Trammels 126 

ElUptical Wheels ; 176. 191 

Elliptical Wheels. Inequality of 192 

Elliott, Professor 94, no, 117 

Engine, Beam 75 

Engine, Cochrane 263 

Engine, Corliss. . 233 

Engine, Direct-acting 99, 159. 261 

Engine, Root ... ........ 263 

Engine, Steering . 241 

English Striking-train 239 

Envelope 183, 213 

Epicyclic Compound Reverted Train 207 

Epicyclic Chamber Wheel-trains 267 

Epicyclic Bevel-gear 328 

Epicyclic Gearing 205 



366 INDEX. 



PAGE 



Epicycloid 197 

Equilateral Curve-triangle 185 

Error due to Obliquity of Connecting-rod 100 

Escape- wheel. 238 

Escapements. 237 

Escapements, Adjustable , 237 

Escapements, Frictional 243, 

Escapements, Periodic 237 

Escapements, Periodical Fluid 273, 

Escapements, Pressure ... 273. 

Escapements, Uniform. 237 

Euler 347 

Ewing's Extensometer 17& 

Example of Angular Velocities in Ball-bearing 340' 

Expansion of Elements 164. 

Extensometer. Ewing 178- 

Feathering Paddle wheel 77 

Field of Restraint 182 

Firing-pin 258 

First Inversion of Slider -crank Chain 99. 

Float-lever of Paddle wheel 79. 

Fluid Escapements 274 

Fluid Links. 24, 264. 

Fluid Links in Screw Mechanisms 282 

Fluid Ratchet-train 264. 

Fly-wheel 173 

Follower 215. 

Follower-pin 220 

Force Closure 140, 171, 173, 221 

Formation of Screw 276 

Forms of Element 1 86 

Forms of Teeth in Screw- and Worm-gearing 293 

Fourth Inversion of Slider-crank Chain 1 20- 

Frame 72 

Free-wheel Bicycle 22& 

Freedom. 19 

Freedom, Degrees of 177 

Friction- brake 232- 

Friction Gearing. . . ... 171, 193 

Frictional Escapements 243. 

Frictional Ratchet 228, 271 

Froude Brake 271 

Function, Periodic 66 

Fusee 255 



INDEX. 367 

PAGE 

Gas-engine, Atkinson's " Cycle " 160 

Gas-meter 274 

Gauge, Micrometer 20 

Gauge, Tide 66 

Gear, Belt-shifting 220 

Gear, Bevel 317 

Gear of Bicycle, Driving 208 

Gear-cutting Machine, Bilgram 323 

Gear-cutting Machine, Rice 325 

Gear, Reversing 242, 281 

Gear, Steering 280 

Gear, Sun-and-planet 209 

Gear-trains 191 

Gear, Two-speed 330 

Gear, Valve 264 

Gear, Winding 252 

Gearing, Epicyclic 205 

Gearing, Friction 171, 193 

Gearing, Ratchet 227 

Gearing, Spur 193 

Gearing, Wheel 150 

Gearing, Worm 285 

Globoidal Cam 226 

Globoidal Screw 285 

Gorge Circle 300 

Governor 234, 273 

Graham's Escapement 238 

Guide-blade 283 

Guide-pulleys 255 

Gun-lock 257 

Gun, Rifled 282 

Head, Rudder 281 

Hearson, Classification of 354 

Helical Pitch 288 

HeHcal Surface 277 

HeHx 295 

Hersey Water-meter 267 

Higher Pairs 5, 21, 23, 168, 183 

Higher Pairing in Spheric Mechanisms 317 

Hindley Worm 298 

Hob 298 

Hole, Slot, and Plane 178 

Hooke's Joint 25, 312 

Hooke's Joint, Angular Velocity in 316 

Hydraulic Accumulator 269 



3^^ INDEX. 

PAGE 

Hydraulic Machines 259 

Hydraulic Press 260 

Hydraulic Transmission 261 

Hyperbola . 171 

Hyperboloid 294 

Hyperboloidal Wheels 298 

Hyperboloidai Wheels. Velocity Ratio in 301 

Hypocycloid 197 

Identity of Mechanisms, Kinematic 117 

Images, Acceleration 155 

Image, Velocity 147, 152 

Incomplete Chains 172 

Incomplete Pairs 171 

Indicator, Richards 87 

Indicator, Steam-engine 167 

Idle Wheels 201 , 207 

Inequality of Elliptical Wheels 192 

Inequality of Lobed Wheels 192 

Instant 29 

Instantaneous Motion 11 

Instantaneous Velocity , . 29 

Internal Bevel-wheels 319 

Inversion 6 

Inversion of a Chain 9 

Inversions of Quadric Crank-chain , 76 

Inversions of Slider-crank Chain 97, 122 

Involute 194, 248 

Involute Cam 219 

Involute Teeth in Bevel -wheels 321 

Involute Wheel-teeth 195 

Joint, Ball-and-socket 21 

Joint, Hooke's 25, 312 

Jon val Turbine 283 

Journal 171 

Kinematic Chain 6 

Kinematic Clamp , 180 

Kinematic Classification 346 

Kinematic Identity of Mechanisms • 117 

Kinematic Link 6 

Kinematic SHde 180 

Kisch's Construction 108 

Kite 83 

Klein's Construction 108 

Knot 27, 35 



INDEX. 369 

PAGE 

Lag 64 

Lanz 34y 

Lathe, Back-gear of 204 

Leakage 260 

Length of Belt 246 

Leonardo da Vinci 130 

Leupold 347 

Lever 71, 216 

Levers, Bell-crank 220 

Lever Crank-chain 77 

Lever, Double 80 

Lever-lock 230 

Linear Velocity 27 

Links, Fluid . . . , 24, 264 

Links. Kinematic 6 

Links, Non-rigid 24 

Links, Tension 244 

Lobed Wheels 191 

Lobed Wheels, Inequality of 192 

Lock, Gun 257 

Lock, Lever 230 

Lock, Yale 232 

Locking-ratchet 230 

Locomotive 82, 174 

Locomotive, Connecting-rod of 82 

Locomotive, Coupling-rod of 82 

Log, Patent 283 

Lower Pair 5, 21 

Lower Pairing in Spheric Mechanisms 308 

MacCord, Professor 302 

Machine \ i, 2 

Machine Design 9 

Machines, Hydraulic 259 

Machine, Screw-making 221 

Machines, Simple 2 

Mechanical Stoker 283 

Machines, Theory of i 

Mechanisms, Alteration of 164 

Mechanisms, Augmentation of 166 

Mechanisms, Classification of 25 

Mechanisms, Conic 309 

Mechanisms, DupUcation of 172 

Mechanisms, Kinematic Identity of 117 

Mechanisms, Order of 26 

Mechanisms, Plane. ...••• • 23 



370 INDEX, 

PAGE 

Mechanisms, Reduced i68 

Mechanisms, Running 264 

Meter, Gas 274 

Meter, Water 274 

Micrometer Gauge 20 

Mill, Wind 283 

Monge , 347 

Motion, Constrained 3 

Motion, Instantaneous 11 

Motion, Non-plane 17 

Motion, Periodic 53. 66, 213 

Motion, Plane 10 

Motion, Quick-return 114 

Motion, Screw 276 

Motion, Simple Harmonic 58 

Motion, Spheric 18 

Motions, Straight-line , 84, 87 

Non-parallel Axes, Belt-gearing between 254 

Non-parallel Axes, Rope-gearing between 254 

jVon-plane Motion 17 

Non-rigid Links 24, 244 

Non-rigid Links in Ratchet-trains 268 

Non-rotative Steam-pumps 168 

Normal Pitch 288 

Nut 20, 277 

Obliquity of Connecting-rod 100 

Obliquity of Connecting-rod, Error Due to 100 

Oldham's Coupling 127 

Open Belt 245 

Order of Mechanisms 26 

Organs, Pressure 264 

Oscillating Engine 112, 262 

Oscillating Engine, Angular Velocity of Cylinder in 112 

Paddle-wheel, Eccentric-pin of 79 

Paddle-wheel, Feathering 77 

Paddle-wheel, Float-levers of 79 

Paddle-wheel, Radius-rods of 79 

Pair 3 

Pair, Cam 213 

Pair-closure 213, 221 

Pair-closure of Chains 175 

Pairing, Higher 23, 168 

Pairing, Pressure 260 



muEx. 371 

PAGE 

Pairs, Incomplete 171 

Pallet 239 

Pantagraph 84 

Pantagraph, Skew 85 

Pappenheim Pump 267 

Parallel Cranks 80 

Parallel-crank Mechanism 174 

Parallel-flow Tm-bine 283 

Parallelogram of Velocities 37 

Patent Log 283 

Pawl 277 

Peaucellier Cell 90 

PeaucelHer Straight-line Motion 90 

Pedestal 140 

Pendulum 67, 238 

Pendulum Pump 1 20 

Period of Simple Harmonic Motion 59 

Periodic Escapements 237 

Periodic Function 66 

Periodic Motion 53, 66, 213 

Periodical Fluid Escapements 273 

Permanent Centre 189 

Phase 64 

Pin, Firing 258 

Pin, Follower 220 

Piston, Acceleration of 102 

Piston Velocity in Direct-acting Engine 100 

Pitch 197, 277 

Pitch-angle 278 

Pitch-angle of Screw-wheels 291 

Pitch, Axial 288 

Pitch-chain 253, 331 

Pitch-circle 193 

Pitch, Circumferential 288 

Pitch, Helical 288 

Pitch, Normal 288 

Pitch-point 193 

Plane Mechanisms 25 

Plane Motion 10 

Planet Wheel 209 

Planing-machine 220 

Plate, Wrist 234 

Plummer-block, Double Adjustment 140 

Point-paths 143 

Point, Pitch 193 

Point of Restraint 177 



372 INDEX. 

PAGE 

Polar Acceleration Diagrams. , , .... ................ . ... . 53, 146, 155, 160 

Polar Velocity Diagrams. ................................. 53, 146, 160 

I*ole 147, 154 

Polygon, Closed .* , 149 

Positive-motion Cam , , 222, 226 

Press, Copying 279 

Press, Hydraulic 260 

Pressure Escapements 273 

Pressure Organs 264 

Pressure Pair , , . , , 24 

Pressiu"e Pairing , , . . , . 260 

Profile of Wheel-teeth 194 

Projectile ^ 282 

Propeller, Screw 283 

Pulley-block, Differential 252 

Pulley-block, Weston Triplex 243 

Pulleys, Cone ^ 246 

Pulleys, Guide 255 

Pump, Centrifugal 264, 285 

Pump, Common 268 

Pump, Differential 269 

Pump, Donkey 121 

Pump, Double-acting , 269 

Pump, Pappenheim 267 

Pump, Pendulum '. 1 20 

Pumps, Steam 125 

Pump, Worthington 132 

Quadric Crank-chain. . . , 70 

Quadric Crank-chain, Angular Velocities in 73 

Quadric Crank-chain, Centrodes of 72 

Quadric Crank-chain, Inversions of . . 76 

Quadric Crank- chain, Virtual Centres of 71 

Quantities, Scalar 35 

Quick-return Motion 114, 118 

Rack 195 

Radial Acceleration 34, 156 

Radial-flow Turbine 285 

Radius-rods of Paddle-wheel 79 

Radius Vector 53 

Rankine. Professor 90, 302, 348 

Rapson's Slide 132 

Ratchet, Checking 230. 271 

Ratchet. Frictional 228, 271 

Ratchet-gearing 227 

Ratchet, Locking 230 



INDEX. 373 

PAGE 

Ratchet, Releasing'. .« 230, 271 

Ratchet, Running. _. >^.. -..„... 277 

Ratchet, Silent ; 229 

Ratchet, Stationary 277 

Ratchet-trains containing Non-rigid Links. . 268 

Ratchet-train, Fluid 264 

Ratchet-wheel 277 

Reduced Centrodes 1 70 

Reduced Mechanisms 168 

Reduction of Chains 168 

Relative Displacement of Bodies having Simple Harmonic Motion 64 

Relative Motion of Bodies having Simple Harmonic Motion 63 

Relative Spinning in Ball-bearings 337 

Releasing-ratchet 230, 27 r 

Renold Chain 254 

Restraint, Field of 182 

Restraint, Point of 177 

Restraining Body i8 f 

Resultant 36 

Resultant Acceleration 37 

Reuleaux, Classification of 35* 

Reuleaux, Professor 70, 167, 170. 183, 186, 237, 247, 262 264. 275, 348 

Reversing-gear , 242, 281 

Reversing-shaft. . 281 

Reverted Compound Epicyclic Train 207 

Reverted Train. 204 

Rice Gear-cutting Machine. 325 

Richards Indicator.. 87 

Rifled Gun. . .". . . 282 

Roberts Straight-line Motion. 90 

Roberval Balance 86 

Roller- bearings. . . . 331 

Rollers, Conical, 331 

Rolling . : 18 333 

Root Biower 267 

Root Engine. 263 

Ropes 24, 244 

Rope-gearing between Non- parallel Axes. 254 

Rope-gearing, Velocity Ratio in . 251 

Rotating Cam. 215 

Rotation, Virtual. 12 

Rudder. . . 241 

Rudder Head. 281 

Ruled Surface r2 

Running Mechanisms. 264 

Running ratchet.. 227 



374 INDEX. 

PAGE 

Safety-valve Spring 259 

Scalar Quantities 35 

Scale of Diagrams 48 

Scott Russell's Straight-line Motion 136 

Screw 20 

Screw Chain, Crossed 281 

Screw, Formation of 276 

Screw-gearing, Forms of Teeth in. 293 

Screw-making Machine 221 

Screw Mechanisms containing Fluid Links 282 

Screw Motion 276 

Screw Pair 22 

Screw-propeller 283 

Screw Surfaces 276 

Screw-thread 277, 278 

Screw-threads, Conical 284 

Screw-threads, Globoidal 285 

Screw-wheels 285 

Screw-wheels, Pitch-angle of 291 

Screw-wheels, Velocity Ratio of 291 

Second Inversion of Slider-crank Chain 112 

Self-centring Chuck 285 

Sense 28 

Setting Out Bevel-wheel Teeth 322 

Shaft, Reversing 281 

Shaping Machine 114 

Shifter, Belt 220 

Shifting-gear, Belt 220 

Silent Ratchet 229 

Simple Chain 7 

Simple Harmonic Motion 58 

Simple Harmonic Motion, Acceleration in 61 

Simple Harmonic Motion, Amplitude of 58 

Simple Harmonic Motion, Composition of 67 

Simple Harmonic Motion, Period of 59 

Simple Harmonic Motion, Relative Motion of Bodies having 63 

Simple Harmonic Motion, Velocity in 60 

Simple Machines 2 

Sine Curve 66 

Skew Bevel- wheels 302 

Skew Pantagraph 85 

Slide Chain, Crossed 131 

Slide, Kinematic 1 80 

Slider Crank 159 

SHder-crank, Crossed Turning 123 

Slider-crank, Crossed Turning-block 123 



INDEX. 375 

PAGE 

Slider-crank, Crossed Swinging 123 

Slider-crank, Crossed Swinging- block , 123 

Slider-crank, Turning-block 263 

Slider-crank, Swinging 120, 263 

Slider-crank, Swinging-block 113, 262 

Slider-crank Chain ; 97, 212 

Slider-crank Chain, Centrodes of 97 

Slider-crank Chain, Crossed •. 122 

Slider-crank Chain, Double 123, 263 

Slider-crank Chain, Inversions of 97, 122 

Slider-crank Chain, Virtual Centres of 97 

Sliding 18 

Sliding Cam 215, 219 

Sliding Pairs, Chain containing only 138 

Snail 241 

Speed 28 

Spheric Mechanisms 25 

Spheric Mechanisms having Higher Pairing 317 

Spheric Mechanisms having Lower Pairing 308 

Spheric Motion 18, 304 

Spheric Triangle 305 

Spinning 18, 333, 342 

Spiral of Archimedes 57, 216, 327 

Spiral Bevel- wheels 327 

Springs 24, 226, 256 

Spring Buffer 259 

Spring Safety-valve 259 

Sprocket-wheel 229. 253, 331 

Spur- gearing 193 

Spur-wheels 193 

Stamp-mill 214 

Stationary Ratchet 277 

Steam-engine 273 

Steam-engine, Brotherhood 173 

Steam-engine, Direct- acting 20, 99, 159, 261 

Steam-engine Indicator 167 

Steam-engine, Oscillating 262 

Steam-engine, Three- cylinder . ........ 173 

Steam- pumps '25 

Steam-pumps, Duplex 132, 168 

Steam-pumps, Non-iotative 168 

Steering- engine 241 

Steering-gear 132, i34- 280 

Steering-wheel 241 

Stoker, Mechanical 283 

Straight-line Motions 84, 87, 136. 167 



376 INDEX. 

PAGE 

Straight-line Motion, Bricard 94 

Straight-line Motion, Cartwright's 212 

Straight-line Motion, Peaucellier 90 

Straight-line Motion, Roberts 90 

Straight-line Motion, Scott Russell's 136 

Straight-line Motion, Tchebicheff 90 

Straight-line Motion, Watt. 88 

Strap-brake .' 272 

Street-car, Diagram of Acceleration of . , ; 46 

Street-car, Diagram of Displacement of 46 

Street-car, Diagram of Velocity of 46 

Striking-train, English 239 

Structure 2, 70 

Sub-normal 52 

Sun-and-planet Gear. 209 

Surface, Helical 277 

Surface, Ruled 12 

Surface, Screw 276 

Swash-plate 226 

Swinging-block Slider-crank 113, 262 

Swinging Slider-crank Chain 120, 263 

Sylvester, Professor 83, 85 

Tangent, Common 288 

Tangential Acceleration . 156 

Tchebicheff Straight- line Motion 90 

Teeth of Bevel-wheels 320 

Teeth, Cycloidal 197 

Teeth, Involute 195 

Tension -links 244 

Tension Pair 24 

Test-piece 178 

Theory of Machines i 

Thickness of Belt 251 

Third Inversion of Slider-crank Chain, 118 

Thread, Screw 277, 278 

Three-cylinder Steam-engine. , 173 

Thrust-bearing. 331, 334 

Tide-gauge 66 

Tiller 133, i35 

Train, Cam 213 

Train, Chamber Crank. 261 

Train, Diagram of Displacement for 43 

Train, Diagram of Velocity for 43 

Trains, Gear 191 

Train, Reverted 204 



INDEX. 377 

PAGE 

Trains, Wheel. ... 7 191 

Trammels, Elliptic 126 

Transformed Centrodes 170 

Transmission, Belt 255 

Transmission, Compressed-air 261 

Transmission, Hydraulic 261 

Tredgold 322 

Triangle, Spheric 305 

Triangle, Vector 149 

Triangular Connecting-rod 175 

Triangle of Velocities 37, 149 

Trigger 258 

Trochoidal Curve 186 

Tumbler 230 

Turbine 264 

Turbine, Jonval 283 

Turbine, Parallel-flow 283 

Turbine, Radial-flow 285 

Turning-block, Slider-crank 263 

Ttuning Pairs 22 

Twist Axis 279, 294 

Two-speed Gear ; 330 

Uniform Acceleration 31, 49, 217 

Uniform Escapements 237 

Uniform Velocity 28,217 

Uniform Velocity Ratio 188 

Universal Joint 312 

Universal Joint, Angular Velocities in 316 

Unwin, Professor 200 

Valve-gear 264 

Valve-gear, Bremme's 143, 151 

Valve-gear, Corliss 233 

Variable Fluid Escapements 274 

Variable Velocity 29 

Variable Velocity Ratio in Belt-gearing 246 

Vector 28, 35 

Vector Addition 36 

Vector, Radius 53 

Vector Triangle 149 

Velocities, Composition of 35 

Velocities, Parallelogram of 37 

Velocities, Triangle of 37, 149 

Velocity 27, 141, 145 

Velocity and Acceleration, Polar Diagrams of 146 



37^ INDEX. 



PAGE 



Velocity, Angular 27, 32 

Velocity, Average 29 

Velocity of Cable-car 53 

Velocity of Crank-pin 102 

Velocity of Cross-head 99 

Velocity, Diagrams of 38, 49 

Velocity Image 147, 152 

Velocity, Instantaneous 29 

Velocity, Linear 27 

Velocity, Magnitude of • 33 

Velocity, Polar Diagrams of 53 

Velocity Ratio in Belt-gearing 244 

Velocity Ratio in Bevel-gear 329 

Velocity Ratio in Bevel-wheels 319 

Velocity Ratio in Cam-trains 222 

Velocity Ratio in Chain-gearing 251 

Velocity Ratio in Hyperboloidal Wheels 301 

Velocity Ratio in Rope-gearing 251 

Velocity Ratio in Screw-wheels 291 

Velocity Ratio, Uniform 188 

Velocity in Simple Harmonic Motion 60 

Velocity, Uniform 28, 217 

Velocity, Variable 29 

Verge 238 

Virtual Axis 12, 305 

Virtual Centre 12 

Virtual Centres in Quadric Crank-chain , 71 

Virtual Centres of Slider-crank Chain 99 

Virtual Rotation 12 

Water-meter. 274 

Water-meter, Hersey •. 267 

Water-wheel 264 

Watt, James 87 

Watt Straight-line Motions 88 

Weighing-machines 171 

Weston Triplex Pulley-block 243. 

Whitworth Quick-return Motion 118 

Wheel, Annular 203, 209 

Wheels, Bevel 306, 318 

Wheels, Double Helical 296 

Wheels, Elliptical 176, 191 

Wheel, Escape 238 

Wheel-gearing 190 

Wheels, Hyperboloidal 298. 

Wheels, Idle 2or 



INDEX. 379 

PAGE 

Wheels, Lobed 191 

Wheel, Planet 209 

Wheel, Ratchet 277 

Wheels, Screw 285 

Wheels, Skew-bevel 302 

Wheel, Sprocket 229, 253, 331 

Wheels, Spur ,. 193 

Wheel, Steering 241 

Wheel-trains 191 

Wheel-trains, Chamber 264 

Wheel-teeth 191 

Wheel-teeth, Cycloidal 197 

Wheel-teeth, Involute 195 

Wheel-teeth, Profiles of 194 

Wheel-trains, Compound 201 

Wheel, Water 264 

Wheel, Worm 286, 290, 297 

Willis, Professor 302, 346 

Windmill 283 

Winding-gear 252 

Worm 290, 297 

Worm-gearing 285 

Worm-gearing, Forms of Teeth in 293 

Worm, Hindley 298 

Worm-wheels 286, 290, 297 

Worthington Pump 132 

Wrist-plate ^ . . 234 

Yale Lock 232 



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1 



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8 



I 



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7 



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** Mechanical Drawing 4to, 4 00 

** Velocity Diagrams Svo, 1 50 

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Reid's Course in Mechanical Drawing Svo, 2 00 

" Text-book of Mechanical Drawing and Elementary Ma- 
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Robinson's Principles of Mechanism Svo, 3 00 

8 



Smith's Manual of Topographical Drawing. (McMillan.) .8vo, 2 50 
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Benjamin's Voltaic Cell Svo, 3 00 

Classen's Qantitative Chemical Analysis by Electrolysis. Her- 

rick and Boltwood.) Svo, 3 00 

Crehore and Squier's Polarizing Photo-chronograph Svo, 3 00 

Dawson's Electric Railways and Tramways.. Small 4to, half mor., 12 60 
Dawson's "Engineering" and Electric Traction Pocket-book. 

16mo, morocco, 4 00 

Slather's Dynamometers, and the Measurement of Power. . 12mo, 3 00 

Gilbert's De Magnete. (Mottelay.) Svo, 2 50 

Holman's Precision of Measurements Svo, 2 00 

" Telescopic Mirror-scale Method, Adjustments, and 

Tests Large Svo, 75 

Landauer's Spectrum Analysis. (Tingle.) Svo, 3 00 

Le Chatelier's High- temperature Measurements. (Boudouard — 

Burgess.) 12mo, 3 00 

Lob's Electrolysis and Electrosynthesis of Organic Compounds. 

(Lorenz.) 12mo, 1 00 

Lyons's Treatise on Electromagnetic Phenomena Svo, 6 00 

• Michie. Elements of Wave Motion Relating to Sound and 

Light Svo, 4 00 

Niaudet's Elementary Treatise on Electric Batteries (Fish- 
back.) 12mo, 2 50 

• Parshall and Hobart's Electric Generators..Small 4to, half mor., 10 00 
Ryan, Norris, and Hoxie's Electrical Machinery. {In preparation.) 
TTiurston's Stationary Steam-engines Svo, 2 50 

• Tillman. Elementary Lessons in Heat Svo, 1 50 

Tory and Pitcher. Manual of Laboratory Physics. .Small Svo, 2 00 

9 



LAW. 

* Davis. Elements of Law 8vo, 2 50 

* " Treatise on the Military Law of United States . . 8vo, 7 00 

* Sheep, 7 50 

Manual for Courts-martial 16mo, morocco, 1 50 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 60 
" Law of Operations Preliminary to Construction in En- 
gineering and Architecture 8vo, 5 00 

Sheep, 5 50 

" Law of Contracts 8vo, 3 00 

Winthrop's Abridgment of Military Law 12mo, 2 59 

MANUFACTURES. 

Beaumont's Woollen and Worsted Cloth Manufacture 12mo, 1 60 

Bemadou's Smokeless Powder — Nitro-cellulose and Theory of 

the Cellulose Molecule l2mo, 2 50 

Holland's Iron Founder 12mo, cloth, 2 50 

" " The Iron Founder " Supplement 12mo, 2 60 

" Encyclopedia of Founding and Dictionary of Foundry 

Terms Used in the Practice of Moulding 12mo, 3 00 

Eissler's Modem High Explosives 8vo, 4 00 

Effront's Enzymes and their Applications. (Prescott.).. .8vo, 3 00 

Fitzgerald's Boston Machinist 18mo, 1 00 

Ford's Boiler Making for Boiler Makers 18mo, 1 00 

Hopkins's Oil-chemists' Handbook 8vo, 3 00 

Keep's Cast Iron 8vo 2 50 

Leach's The Inspection and Analysis of Food with Special 
Reference to State Control. {In preparation.) 

Metcalf's Steel. A Manual for Steel-users 12mo, 2 00 

Metcalfs Cost of Manufactures — And the x^dministration of 

Workshops, Public and Private 8vo, 5 00 

Meyer's Modem Locomotive Construction 4to, 10 00 

* Reisig's Guide to Piece-dyeing 8vo, 25 00 

Smith's Press- working of Metals 8vo, 3 00 

" Wire: Its Use and M&nufacture Small 4to, 3 00 

Spalding's Hydraulic Cement 12mo, 2 00 

Spencer's Handbook for Chemists of Beet-sugar Houses. 

16mo, morocco, 3 00 
" Handbook for Sugar Manufacturers and their Chem- 
ists 16mo, morocco, 2 00 

Thurston's Manual of Steam-boilers, their Designs, Construc- 
tion and Operation 8vo, 5 00 

Walke's Lectures on Explosives 8vo, 4 00 

West's American Foundry Practice 12mo, 2 60 

" Moulder's Text-book 12mo, 2 50 

Wiechmann's Sugar Analysis Small 8vo, 2 60 

Wolflf's Windmill as a Prime Mover 8vo, 3 00 

Woodbury's Fire Protection of Mills 8vo, 2 00 

MATHEMATICS. 

Baker's Elliptic Functions 8vo, 1 60 

* Bass's Elements of Differential Calculus 12mo, 4 00 

Briggs's Elements of Plane Analytic Geometry 12mo, 1 00 

Chapman's Elementary Course in Theory of Equationa. . .12mo, 1 50 

Compton's Manual of Logarithmic Computations 12mo, 1 60 

10 



Davis's Introduction to the Logic of Algebra 8vo, 1 60 

•Dickson's College Algebra Large 12mo, 1 50 

Halsted's Elements of Geometry 8vo, 1 75 

" Elementary Synthetic Geometry 8vo, 1 50 

• Johnson's Three-place Logarithmic Tables: Vest-pocket size, 

pap., 15 

100 copies for 5 00 

• Mounted on heavy cardboard, 8 X 10 inches, 25 

10 copies for 2 00 
" Elementary Treatise on the Integral Calculus. 

Small 8yo, 1 50 

" Curve Tracing in Cartesian Co-ordinates 12mo, 1 00 

" Treatise on Ordinary and Partial Differential 

Equations Small 8vo, 3 50 

" Theory of Errors and the Method of Least 

Squares 12mo, 1 50 

• ** Theoretical Mechanics. , . . 12mo, 3 00 

Laplace's Philosophical Essay on Probabilities. (Truscott and 

Emory.) '. 12mo, 2 GO 

• Ludlow and Bass. Elements of Trigonometry and Logarith- 

mic and Other Tables 8vo, 3 00 

** Trigonometry. Tables published separately. .Each, 2 00 

Merriman and Woodward. Higher Mathematics 8vo, 5 00 

Merriman's Method of Least Squares 8vo, 2 00 

Rice and Johnson's Elementary Treatise on the Dijfferential 

Calculus Small 8vo, 3 00 

" Differential and Integral Calculus. 2 vols. 

in one Small 8vo, 2 50 

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" Trigometry: Analytical, Plane, and Spherical 12mo, 1 00 



MECHANICAL ENGINEERING. 

MATERIALS OF ENGINEERING, STEAM ENGINES 
AND BOILERS. 

Baldwin's Steam Heating for Buildings 12mo, 2 60 

Barr's Kinematics of Machinery 8vo, 2 50 

• Bartlett's Mechanical Drawing 8vo, 3 00 

Benjamin's Wrinkles and Recipes 12mo, 2 00 

Carpenter's Experimental Engineering. 8vo, 6 00 

'^ Heating and Ventilating Buildings 8vo, 4 00 

Clerk's Gas and Oil Engine Small 8vo, 4 00 

Coolidge's Manual of Drawing 8vo, paper, 1 GO 

Cromwell's Treatise on Toothed Gearing 12mo, 1 50 

" Treatise on Belts and Pulleys 12mo, 1 50 

Durley's Mementary Text-book of the Kinematics of Machines. 

{In preparation.) 

Mather's Dynamometers, and the Measurement of Power . . 12mo, 3 00 

" Rope Driving 12mo, 2 00 

Gill's Gas an Fuel Analysis for Engineers 12mo, 1 26 

Hall's Car Lubrication 12mo, 1 00 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 1 50 

Part 11. — Form, Strength and Proportions of Parts 8vo, 3 00 

Kent's Mechanical Engineers' Pocket-book 16mo, morocco, 5 00 

Kerr's Power and Power Transmission 8vo, 2 00 



MacCord's Kinematics; or. Practical Mechanism 8vo, 5 00 

" Mechanical Drawing 4to, 4 00 

" Velocity Diagrams 8vo, 1 50 

Mahan's Industrial Drawing. (Thompson.) 8vo, 3 50 

Poole's Calorific Power of Fuels Svo, 3 00 

Raid's Course in Mechanical Drawing Svo, 2 00 

" Text-book of Mechanical Drawing and Elementary 

Machine Design Svo, 3 00 

Richards's Compressed Air 12mo, 1 50 

Robinson's Principles of Mechanism ,Svo, 3 00 

Smith's Press-working of Metals Svo, 3 00 

Thurston's Treatise on Friction and Lost Work in Machin- 
ery and Mill Work Svo, 3 00 

** Animal as a Machine and Prime Motor and the 

Laws of Energetics 12mo, 1 00 

Warren's Elements of Machine Construction and Drawing. .Svo, 7 60 
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mann—Klein.) Svo, 5 00 

" Machinery of Transmission and Gorvemors. (Herr- 
mann—Klein.) Svo, 5 00 

" Hydraulics and Hydraulic Motors. (Du Bois.) .Svo, 5 00 

Wolff's Windmill as a Prime Mover Svo, 3 00 

Wood's Turbines Svo, 2 60 

MATERIALS OF ENGINEERING. 

Bovey's Strength of Materials and Theory of Structures. .Svo, 7 50 
Burr's Elasticity and Resistance of the Materials of Engineer- 
ing Svo, 5 00 

Church's Mechanics of Engineering Svo, 6 00 

Johnson's Materials of Construction Large Svo, 6 00 

Keep's Cast Iron Svo, 2 50 

Lanza's Applied Mechanics Svo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) Svo, 7 50 

Merriman'a Text-book on the Mechanics of Materials. .. .Svo, 4 00 

" Strength of Materials 12mo, 1 00 

Metcalf's Steel. A Manual for Steel-users 12mo, 2 00 

Smith's Wire: Its Use and Manufacture Small 4to, 3 00 

" Materials of Machines 12mo, 1 GO 

Thurston's Materials of Engineering 3 vols., Svo, 8 00 

Part II.— Iron and Steel Svo, 3 50 

Part III. — A Treatise on Brasses, Bronzes and Other Alloys 

and their Constituents Svo, 2 60 

Thurston's Text-book of the Materials of Construction. .. .Svo, 5 00 
Wood's Treatise on the Resistance of Materials and an Ap- 
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" Elements of Analytical Mechanics Svo, 3 00 

STEAM ENGINES AND BOILERS. 

Camot's Reflections on the Motive Power of Heat. (Thurston.) 

12mo, 1 60 
Dawson's " Engineering " and Electric Traction Pocket-book. 

16mo, morocco, 4 00 

Ford's Boiler Making for Boiler Makers ISmo, 1 00 

Goss's Locomotive Sparks Svo, 2 00 

Hemenway's Indicator Practice and Steam-engine Economy. 

12mo, 2 00 

Button's Mechanical Engineering of Power Plants Svo, 5 00 

" Heat and Heat-engines Svo, 6 00 

12 



Kent's Steam-boiler Economy 8vo, 4 00 

Kneass's Practice and Theory of the Injector 8vo, 1 60 

MacCord's Slide-valves 8vo, 2 00 

Meyer's Modem Locomotive Construction 4to, 10 00 

Peabody's Manual of the Steam-engine Indicator 12mo, 1 50 

" Tables of the Properties of Saturated Steam and 

Other Vapors Svo, 1 00 

" Thermodynamics of the Steam-engine and Other 

Heat-engines Svo, 5 00 

** Valve-gears for Steam-engines Svo, 2 50 

Peabody and Miller. Steam-boilers Svo, 4 00 

Pray's Twenty Years with the Indicator Large Svo, 2 50 

Pupin's Thermodynamics of Reversible Cycles in Gases and 

Saturated Vapors. (Osterberg.) 12mo, 1 25 

Reagan's Locomotive Mechanism and Engineering 12mo, 2 00 

Rontgen's Principles of Thermodynamics. (Du Bois.) Svo, 5 00 

Sinclair's Locomotive Engine Running and Management. .12mo, 2 00 

Smart's Handbook of Engineering Laboratory Practice. .12mo, 2 50 

Snow's Steam-boiler Practice Svo, 3 00 

Spangler's Valve-gears Svo, 2 50 

" Notes on Thermodynamics 12mo, 1 00 

Thurston's Handy Tables Svo, 1 50 

" Manual of the Steam-engine 2 vols., Svo, 10 00 

Part I. — ^History, Structure, and Theory Svo, 6 00 

Part II. — Design, Construction, and Operation Svo, 6 00 

Thurston's Handbook of Engine and Boiler Trials, and the Use 

of the Indicator and the Prony Brake Svo, 5 00 

" Stationary Steam-engines Svo, 2 50 

" Steam-boiler Explosions in Theory and in Prac- 
tice 12mo, 1 50 

" Manual of Steam-boilers, Their Designs, Construc- 
tion, and Operation Svo, 5 00 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.).. Svo, 5 00 

Whitham's Steam-engine Design Svo, 5 00 

Wilson's Treatise on Steam-boilers. (Flather.) 16mo, 2 50 

Wood's Thermodynamics, Heat Motors, and Refrigerating 

Machines Svo, 4 00 

MECHANICS AND MACHINERY. 

Barr's Kinematics of Machinery Svo, 2 50 

Bovey's Strength of Materials and Theory of Structures. .Svo, 7 50 

Chordal. — ^Extracts from Letters. 12mo, 2 00 

Church's Mechanics of Engineering Svo, 6 00 

" Notes and Examples in Mechanics Svo, 2 00 

Compton's First Lessons in Metal-working 12mo, 1 50 

Compton and De Groodt. The Speed Lathe. 12mo, 1 50 

Cromwell's Treatise on Toothed Gearing 12mo, 1 50 

" Treatise on Belts and Pulleys 12mo, 1 50 

Dana's Text-book of Elementary Mechanics for the Use of 

Colleges and Schools 12mo, 1 50 

Dingey's Machinery Pattern Making 12mo, 2 00 

Dredge's Record of the Transportation Exhibits Building of the 

World's Columbian Exposition of 1893 4to, h^f mor., 6 00 

Du Bois's Elementary Principles of Mechanics: 

Vol. I. — Kinematics Svo, 3 50 

Vol. II.— Statics Svo, 4 00 

Vol. HI.— Kinetics Svo, 3 50 

Du Bois's Mechanics of Engineering. Vol. I Small 4to, 7 50 

« « « « " Vol.11 Small 4to, 10 00 

13 



DurleT^s Elementary Text-book of the Kinematics of Machines. 

{In preparation.) 

Fitzgerald's Boston Machinist 16mo, 1 00 

Mather's Dynamometers, and the Measurement of Power. 12mo, 3 00 

" Rope Driving 12mo, 2 00 

Goss's Locomotive Sparks 8vo, 2 00 

Hall's Car Lubrication 12mo, 1 00 

Holly's Art of Saw Filing 18mo, 75 

♦ Johnson's Theoretical Mechanics 12mo, 3 00 

Johnson's Short Course in Statics by Graphic and Algebraic 

Methods. {In preparation.) 
Jones's Machine Design: 

Part I- — Kinematics of Machinery 8vo, 1 60 

Part II. — Form, Strength and Proportions of Parts. .. .Svo, 3 00 

Kerr's Power and Power Transmission Svo, 2 00 

Lanza's Applied Mechanics Svo, 7 50 

MacCord's Kinematics; or, Practical Mechanism Svo, 5 00 

" Velocity Diagrams Svo, 1 50 

Merriman's Text-book on the Mechanics of Materials Svo, 4 00 

• Michie's Elements of Analytical Mechanics Svo, 4 00 

Reagan's Locomotive Mechanism and Engineering 12mo, 2 00 

Eeid's Course in Mechanical Drawing Svo, 2 00 

" Text-book of Mechanical Drawing and Elementary 

Machine Design Svo, 3 00 

Richards's Compressed Air 12mo, 1 50 

Robinson's Principles of Mechanism Svo, 3 00 

Ryan, Norris, and Hoxie's Electrical Machinery. {In preparation.) 

Sinclair's Locomotive-engine Running and Management. . 12mo, 2 00 

Smith's Press-working of Metals Svo, 8 00 

" Materials of Machines 12mo, 1 00 

Thurston's Treatise on Friction and Lost Work in Machin- 
ery and Mill Work Svo, 3 00 

" Animal as a Machine and Prime Motor, and the 

Laws of Energetics 12mo, 1 00 

Warren'e Elements of Machine Construction and Drawing. .Svo, 7 60 
Weisbach's Kinematics and the Power of Transmission. 

(Herrman — Klein.) Svo, 5 00 

" Machinery of Transmission and Governors. (Herr- 

(man — Klein.) Svo, 6 00 

Wood's Elements of Analytical Mechanics Svo, 3 00 

" Principles of Elementary Mechanics 12mo, 1 26 

" Turbines Svo, 2 50 

The World's Columbian Exposition of 1S93 4to, I 00 

METALLURGY. 

Egleston's Metallurgy of Silver, Gold, and Mercury: 

Vol. L— Silver Svo, 7 50 

Vol. n.— Gold and Mercury Svo, 7 60 

** Hes's Lead-smelting 12mo, 2 50 

Keep's Cast Iron Svo, 2 50 

Kunhardt's Practice of Ore Dressing in Ilirope Svo, 1 50 

Le Chatelier's High-temperature Measurements. (Boudouard — 

Burgess.) 12mo, 3 00 

Metcalfe Steel. A Manual for Steel-users 12mo, 2 00 

Smith's Materials of Machuies 12mo, 1 00 

Thurston's Materials of Engineering. In Three Parts Svo, S 00 

Part II. — Iron and Steel Svo, 3 60 

Part in. — ^A Treatise on Brasses, Bronzes and Other Alloys 

and Their Constituents Svo, 2 60 

14 



MINERALOGY. 

Barringer's Description of Minerals of Commercial Value. 

Oblong, morocco, 2 60 

Boyd's Resources of Southwest Virginia 8vo, 3 00 

" Map of Southwest Virginia Pocket-book form, 2 00 

Brush's Manual of Determinative Mineralogy. (Penfield.) .8vo, 4 00 

Chester's Catalogue of Minerals 8vo, paper, 1 00 

Cloth, 1 26 

" Dictionary of the Names of Minerals Svo, 3 60 

Dana's System of Mineralogy Large Svo, half leather, 12 60 

" First Appendix to Dana's New " System of Mineralogy." 

Large 8vo, 1 OU 

** Text-book of Mineralogy 8vo, 4 00 

** Minerals and How to Study Them 12ma, 1 50 

" Catalogue of American Localities of Minerals . Large 8vo, 1 00 

** Manual of Minera.logy and Petrography 12mo, 2 00 

Ejgleston's Catalogue of Minerals and Synonyms 8vo, 2 60 

Hussak's The Determination of Rock-forming Minerals. 

(Smith.) Small 8vo, 2 00 

• Penfield 's Notes on Determinative Mineralogy and Record of 

Mineral Tests 8vo, paper, 60 

Eosenbusch's Microscopical Physiography of the Rock-making 

Minerals. (Idding's.) 8vo, 5 00 

•Tillman's Text-book of Important Minerals and Rocks.. 8vo, 2 00 

Williams's Manual of Lithology Svo, 3 00 



MINING. 

Beard's Ventilation of Mines 12mo, 2 60 

Boyd's Resources of Southwest Virginia Svo, 3 00 

** Map of Southwest Virginia Pocket-book form, 2 00 

•Drinker's Tunneling, Explosive Compounds, and Rock 

Drills 4to, half morocco, 25 00 

Eissler's Modern High Explosives Svo, 4 00 

Fowler's Sewage Works Analyses 12mo, 2 00 

Groodyear's Coal-mines of the Western Coast of the United 

States 12mo, 2 50 

Dilseng's Manual of Mining Svo, 4 00 

♦* Iles's Lead-smelting 12mo, 2 50 

Kunhardt's Practice of Ore Dressing in Europe Svo, 1 50 

O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 00 

Sawyer's Accidents in Mines Svo, 7 00 

Walke's Lectures on Explosives Svo, 4 00 

Wilson's Cyanide Processes 12mo, 1 50 

Wilson's C^lorination Process 12mo, 1 60 

Wilson's Hydraulic and Placer Mining 12mo, 2 00 

Wilson's Treatise on Practical and Theoretical Mine Ventila- 
tion 12mo, 1 26 

SANITARY SCIENCE. 

Folwell's Sewerage. (Designing, Construction and Maintenance.) 

Svo, 3 00 

" Water-supply Engineering Svo, 4 00 

Fuertes's Water and Public Health 12mo, 1 60 

" Water-filtration Works 12mo, 2 60 

15 



Gerhard's Guide to Sanitary House-inspection 16mo, 1 00 

Goodrich's Economical Disposal of Towns* Refuse. . .Demy 8vo, 3 60 

Hazen's Filtration of Public Water-supplies 8vo, 3 00 

Kiersted's Sewage Disposal „ 12mo, 1 26 

Leach's The Inspection and Analysis of Food with Special 

Reference to State Control. {In preparation.) 
Mason's Water-supply. (Considered Principally from a San- 
itary Standpoint. 3d Edition, Rewritten 8vo, 4 00 

** Examination of Water. (Chemical and Bacterio- 
logical.) 12mo, 1 26 

Merriman's Elements of Sanitary Engineering Svo, 2 00 

Nichols's Water-supply. (Considered Mainly from a Chemical 

and Sanitary Standpoint.) ( 1883.) 8vo, 2 60 

Ogden's Sewer Design 12mo, 2 00 

• I*rice's Handbook on Sanitation 12mo, 1 60 

Kichards's Cost of Food. A Study in Dietaries 12mo, 1 00 

Richards and Woodman's Air, Water, and Food from a Sani- 
tary Standpoint 8vo, 2 00 

Richards's Cost of Living as Modified by Sanitary Science. 12mo, 1 00 

* Richards and Williams's The Dietary Computer Svo, 1 60 

Rideal's Sewage and Bacterial Purification of Sewage Svo, 3 60 

Tumeaure and Russell's Public Water-supplies 8vo, 6 00 

Whipple's Microscopy of Drinking-water 8vo, 3 60 

WoodhuU's Notes on Military Hygiene 16mo, 1 60 



MISCELLANEOUS. 

Barker's Deep-sea Soundings 8vo, 2 00 

Emmoiis's Geological Guide-book of the Rocky Mountain Ex- 
cursion of the International Congress of Geologists. 

Large 8vo, 1 60 

Ferrel's Popular Treatise on the Winds 8vo, 4 00 

Haines's American Railway Management 12mo, 2 60 

Mott's Composition, Digestibility, and Nutritive Value of Food. 

Mounted chart, 1 26 

" Fallacy of the Present Theory of Sound 16mo, 1 00 

Ricketts's History of Rensselaer Polytechnic Institute, 1824- 

1894 Small 8vo, 3 00 

Rotherham's Emphasised New Testament Large Svo, 2 Of^ 

" Critical Emphasised New Testament 12mo, 1 60 

Steel's Treatise on the Diseases of the Dog Svo, 3 60 

Totten's Important Question in Metrology Svo, 2 60^ 

The World's Columbian Exposition of 1893 4to, 1 00 

Worcester and Atkinson. Small Hospitals, Establishment and 
Maintenance, and Suggestions for Hospital Architecture, 

with Plans for a Small Hospital 12mo, 1 26 



HEBREW AND CHALDEE TEXT-BOOKS. 

Green's Grammar of the Hebrew Language Svo, 3 00 

" Elementary Hebrew Grammar 12mo, 1 26 

" Hebrew Chrestomathy Svo, 2 00 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament 

Scriptures. (Tregelles.) Small 4to, half morocco, 6 00- 

Letteris's Hebrew Bible Svo, 2 26- 

16 



FEB 4 1903 



)903 



